Modelling and Calculus

Transcription

1 Moelling an Calculus By Collin Phillips This material has been evelope as a joint project between the Mathematics Learning Centre MLC an the Learning Centre LC at the University of Syney. With literary an structural contributions from Dr Janet Jones an Ms Helen Drury of the LC. Thanks for the many useful suggestions an corrections by Dr Sue Goron an Ms Jackie Nicholas of the MLC an, Mr George Papaopoulos an Prof. Leon Polaian of the School of Mathematics an Statistics. Thanks also for the feeback an proofreaing of the many stuents of the MLC incluing Dr Rukshana Yates an Ms Vanessa Kung. This material was evelope with the ai of a Teaching Improvement an Equipment Scheme TIES Grant at the University of Syney in

2 Contents 1 Moelling an Calculus 1 MAC 1 The interpretation an translation of natural an real worl problems that are escribe in wors into a specification an escription of the moelling problem in the language of mathematics Rates of Change an Derivatives Ientifying The Difference Between a Quantity an the Rate of Change of That Quantity Exercises What is a Differential Equation? Description of a Differential Equation Constant Rate of Change Exercises Proportionality Description of Proportional Quantities Exercises Constants of Proportionality Exercises Moelling an Calculus 2 MAC 2 Unerstaning the concepts an ieas of ifferential equations an their solutions in terms of written wor escriptions of the ifferential equations an the concepts of solving a ifferential equation Differential Equations as Questions: Various Variables Solution Function an Inepenent Variable Inepenent Variables an Depenent Variables Alternative Notations for Differential Equations Exercises Differential Equations as Questions Differential Equations of the Form y = f(x) as Questions x Exercises Differential Equations of the Form y = g(y) as Questions x Differential Equations of the Form y = f(x)g(y) as Questions x Exercises Particular Solutions an General Solutions or Differentiating in Reverse The ifference between General Solutions an a Particular Solution Exercises Particular Solutions an General Solutions in General Exercises

3 3 Moelling an Calculus 3 MAC 3 Solving some ifferential equations by using the concepts an interpretations of a ifferential equation to fin a solution or an answer pose by the ifferential equation Solving Differential Equations of the Form y = f(x) Using Differentiation in Reverse x Exercises Differentiation Tables Solving Differential Equations of the Form y = f(x) Using Integration x Exercises Checking Your Solution Exercises Integration Tables Solving Differential Equations of the Form y = f(y) x Differential Equation of the Form y x = y Exercises Differential Equation of the Form y = ky x Exercises How Not to Solve a Differential Equation of the Form y = f(y) x 4 Moelling an Calculus 4 MAC 4 Relating written wor escriptions of real worl physical conitions to the escription of the problem in the language of mathematics an using these escriptions to fin solutions to the physical problem Fining Conitions in Questions Exercises Ten Important Steps for Solving Moelling Questions Pose in Wors Answers to Selecte Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises Answers to Exercises

4 Chapter 1 Moelling an Calculus 1 MAC 1 The interpretation an translation of natural an real worl problems that are escribe in wors into a specification an escription of the moelling problem in the language of mathematics 1.1 Rates of Change an Derivatives Ientifying The Difference Between a Quantity an the Rate of Change of That Quantity Imagine a block of ice put in one of your classrooms. The block of ice will melt. The warmer the room the quicker the ice will melt. How o we write this as an equation? Let us say the volume of the block of ice is given by V. Then the rate at which the block will melt will be V, where t is the time. t Remember the t represents how quickly something changes. The rate of change of volume, in wors, is represente by V t in mathematics. Now the temperature of the room influences how quickly the ice will melt. The temperature oes not change the volume of ice instantaneously. A hot room oesn t mean there will be no ice straight away. So the temperature of the room influence V t not V irectly. 4

5 Draft Version August 2011 c 2010 University of Syney 5 In nature an the physical worl often one quantity will influence the rate of change of another. Here we are making the istinction between a quantity an how quickly that quantity changes. These are not inter-changable concepts. There may be lots of ice in a freezer. In which case the volume is large but the ice is changing very slowly. There may be an iceberg which has rifte to a location with a warm climate in which case there is lots of ice changing very quickly. There may be an ice cube in a col slushy rink in which case there is little ice changing slowly. There may be an ice cube in a warm beer in which case there is little ice that will change rapily. This istinction between the amount or quantity of a substance, an the rate of change of that substance is a critical concept for unerstaning moelling an calculus. In written wors expressions like rate, rate of change, spee, acceleration, how quickly, how slowly, how fast, amongst others may inicate a erivative in mathematics Exercises In the following sentences ientify which part, or parts, of the sentence represents x or the erivative of a t quantity. For each question you nee to make a istinction between which part of the question escribes the quantity of something in the problem an which parts escribe the rate of change of that quantity in the problem; just as we mae a istinction between the volume of ice an the rate at which the volume of ice changes in the examples above. 1. A col sausage is place in an oven. The rate of increase of temperature of the sausage will epen on how hot the oven is. 2. The rate of change of concentration of salt in a cell will epen on the ifference between the concentration of salt in the cell an in the concentration of salt in the environment. 3. The rate at which a human boy prouces insulin will epen on the concentration of sugars in the bloo 4. How fast a car travels will epen on the rate at which fuel is being taken from the fuel tank an fe to the engine. 5. How fast a car accelerates or changes velocity epens on the rate of change of the rate at which fuel is being taken from the fuel tank an fe to the engine. 6. A swimming pool, which is initially full of water, is raine through a hole in the bottom of the pool. The rate at which the epth of water rops will epen on the the pressure at the bottom of the pool an hence will epen on the epth of water the water in the pool. 7. A ballon is fille with air an then allowe to eflate. The larger the balloon the more pressure the air will exert. The rate of change of the volume of the balloon will epen on the iameter of the balloon.

6 Draft Version August 2011 c 2010 University of Syney What is a Differential Equation? Description of a Differential Equation In nature, very often one property of a system will influence the rate of change of another property. For instance if we place a hot pie in a warm room the rate of change of the temperature of the pie will epen on how col or warm the room is. If the pie is put in a freezer the pie will cool quickly. If the pie is put in a warm room it will cool more slowly. If place in an oven it will cool slowly or even heat up, epening on how hot the oven is, an in this case how hot the pie is. The rate of change of the temperature of the pie will epen on the temperature of the room. Many, many physical, chemical, electrical, biological an other natural systems can be well moele by relationships between one property of the system an the rate of change of another. Many, many physical, chemical, electrical, biological an other natural systems can be well moele by relationships between one property of the system an the rate of change of another. For these reasons we nee to incorporate erivatives into our equations. A ifferential equation is an equation which involves a erivative of one of the variables Constant Rate of Change If we put a hose in a swimming pool an turn the tap on full the pool will fill up. If the tap elivers water at, let s say, 1000 litres every hour an this rate oesn t change then the volume of water in the pool will change by a certain amount in any given time perio. If the volume of water in the pool is V then the rate of change of V will be constant. In mathematics this iea is simply written as: where c is a constant an t is time. V t = c, This mathematical expression has an equal sign hence it is calle an equation. The equation also involve a erivative V hence it is calle a ifferential equation. t The expression V t = c, is an equation an has a erivative of one of the variables; hence we call it a ifferential equation. If a pool initially has 2000 litres of water in it an then has 3000 litres of water after constantly filling for 1 hour an 4000 litres of water after 2 hours an 6000 litres after 4 hours then V t = 1000 litres hour, or, for V measure in litres an t measure in hours we just write V t = Here the change in volume for any fixe perio of time is the same. The volume changes by 1000 in any hour. The rate of change in volume will epen on how much we turn the tap on. If we turn-off the tap a bit the rate will ecrease, but the volume will not. The volume of water in the pool is ifferent than the rate of increase of volume. The rate at which the water enters the pool may be 1000 litres per hour, but the volume in the pool is never 1000 litres.

7 Draft Version August 2011 c 2010 University of Syney Exercises For the following wore questions ientify which parts of the question represents the erivative of a quantity, which part represents the quantity itself, which part represents the equality, an which feature of the problems tell us what the rate of change will be. 1. A ballon fills with air such that the rate of change of volume of the balloon is equal to 2000 litres per minute. 2. The rate of change of temperature of a col beer is equal to the ifference in temperature of the beer an the room. 3. The rate at which a swimming pool is fille is equal to a constant times the opening in the tap. 4. A bowling ball is thrown out of an aircraft an falls such that the rate of change of the spee is increasing by 9.8 metres per secon every secon. (This means that the object, which initially is not falling at all will be falling at 9.8 metres per secon after 1 secon, an will be falling at 19.6 metres per secon after 2 secons etcetera.) 5. A balloon eflates such that the rate of change of volume of the balloon is equal to a constant times the volume of the balloon. In this case as the balloon eflates there is less pressure exerte by the balloon on the air insie. 6. The rate of change of concentration of alcohol in the bloo stream of a person who has stoppe rinking is constant. 7. The change of spee of an object falling to Earth will increase at a constant rate of 9.8 metres per secon every secon. (This means that if the object is roppe from rest then it will be falling at a spee of 9.8 metres per secon after one secon an then be falling at a spee of 19.6 metres per secon after 2 secons.) 8. The rate of change of the win-spee (where the win-spee is positive if it is onshore an negative if is or offshore) in an iealize costal region will be epenent on the ifference in the temperature of the lan an the ocean. 9. The rate at which a small shark population increases will equal a constant times the number of fish in their habitat

8 Draft Version August 2011 c 2010 University of Syney Proportionality Description of Proportional Quantities The iameter of a circle is always two times the raius. For a circle of raius 1 metre the iameter is 2. For a circle of raius 1 light year the iameter is 2 light years. A light year is how far light travels in a year (in a vacuum). This is a pretty big circle. Likewise there is a relationship between the istance aroun a circle (the circumference) an the iameter. For a tractor wheel of iameter of 1 metre the circumference is about metres = π m. For a ferris wheel of iameter of 10 metres the circumference is about metres = 10π m. If a fixe change in one quantity always leas to, a not necessarily equal, but fixe change in another we say that the two quantities are proportional. For instance if we change the raius of a circle by 1 metre then we will change the iameter by 2 metres. If we increase the iameter of a circle by 1 metre then we will change the circumference by m or π m. We say that the iameter is proportional to the raius or in mathematical symbols we write iameter raius or r, where is the iameter of the circle an r is the raius. This means that the iameter will be equal to a constant times the raius. Here =2r. Likewise the circumference is proportional to the iameter of a circle or circumference iameter or c, where c is the circumference. Here the circumference is equal to a constant times the iameter, an the constant is the most famous constant of proportionality, π. We write c = π If two quantities A an B are proportional we write A B, which reas A is proportional to B. If A an B are proportional then A will be equal to a non-zero constant times B or A = k B, which reas A is equal to a non zero constant k times B.

9 Draft Version August 2011 c 2010 University of Syney Exercises Are the following quantities proportional? 1. The iameter of a sphere an the raius of a sphere. 2. The circumference of a circle an the iameter of a circle. 3. The circumference of a great circle (the largest circumference of a sphere) an the iameter of a sphere. 4. The perimeter of a square an the length of a sie of a square. 5. The area of a square an the sie length of a square. 6. The area of a circle an the raius of a circle. 7. The height an the corresponing weight of each person in a group of 100 stuents. 8. The number of people passing a maths class an the size of that maths. 9. The area of a circle an the raius-square of the circle. 10. The volume of a cube an the volume of the biggest sphere that can just be containe in that sphere Constants of Proportionality If A is proportional to B then A = k B, where k 0 is calle the constant of proportionality. It is generally specifie that the constant of proportionality not be equal to zero. If a constant c coul be zero then A = c B woul mean that A = 0 an A woul be ientically equal to zero, no matter what the value of B. For this circumstance the value of A woul not be epenent on B or the two quantities A an B woul not be epenent. Since we reasonably expect two quantities that are proportional to be epenent we generally exclue the case of k = 0 from the efinition. Perhaps the most famous constant of proportionality is the constant π, which is about We know the iameter an circumference of a circle c are proportional, or c in this case c = π. So π is the constant of proportionality between the iameter an circumference of a circle. In fact this can be use as a efinition of the important constant π. π is the geometric constant of proportionality (for Eucliean space). For a fixe change of say 1 metre in iameter, the circumference will change by or π metres. If we increase the iameter of a Ferris wheel from 10 metres to 11 meters it will be metres larger aroun the circumference. In fact if we change the iameter of any Ferris wheel by 1 metre, no matter how big or small, the circumference will change by π metres. This is precisely why, when you pump up the tyres on a car or a bike the speeo will rea slower when you are going at exactly the same spee along the roa. Or another way of saying this is that for the same reaing on the speeo more pumpe up tyres will make the car travel faster. If we graph the iameter of a ferris wheel with the circumference the graph will be a straight line. Every time we increase by one metre c will increase by π metres.

10 Draft Version August 2011 c 2010 University of Syney 10 The graph of any two proportional quantities with A on the horizontal axis B on the vertical axis, with say A = k B will be a straight line with graient k. If we change A by A this will lea to a change in B of k B Exercises For each of the following sets of quantities fin the constant of proportionality 1. The raius of a sphere an the iameter of a sphere. 2. The iameter of a sphere an the circumference of a great circle ( the largest circumference of a sphere). 3. The length of a sie of a square an the istance aroun the perimeter of a square. 4. The raius of a circle an the circumference of a circle. 5. The volume of a cube an the sie length cube of the cube. 6. The volume of a sphere an the raius of a sphere cube. 7. An inch an a millimetre. 8. A litre an a cubic metre. 9. The sie length of a square an the iagonal of a square. 10. The area of a square an the area of the largest circle that can wholly be containe in that square. 11. The volume of a cube an the volume of the biggest sphere that can wholly be containe in that cube. Hint: for half the sie length of the cube an r the raius of the sphere then r 2 = What constant of proportionality woul you choose for the following quantity. The number of people passing a maths class an the size of that maths class. 13. What constant of proportionality woul you choose for the following quantity. The number of people failing a maths class an the size of that maths class. 14. The area of a circle an the raius square of the circle.

11 Chapter 2 Moelling an Calculus 2 MAC 2 Unerstaning the concepts an ieas of ifferential equations an their solutions in terms of written wor escriptions of the ifferential equations an the concepts of solving a ifferential equation. 2.1 Differential Equations as Questions: Various Variables Solution Function an Inepenent Variable To work out what question a ifferential equation is asking we look at the symbol at the top of the erivative, for y f this is y. For this is f. We may want to eventually fin this function. Let s call this the solution x t function. We then look at the thing we are ifferentiating with respect to, or the symbol on the bottom of the erivative. For y f this is x. For this is t. Here we will call this the inepenent variable, because (here) we want to x t fin the solution function in terms of this inepenent variable. Here we want to fin the solution function in terms of the inepenent variable. For y we want to fin y in terms of x. x For f we want to fin f in terms of t. t Inepenent Variables an Depenent Variables The reason that we call one variable inepenent an one variable a epenent variable is that they will serve ifferent rolls in the solution. 11

12 Draft Version August 2011 c 2010 University of Syney 12 For instance if we fin y in terms of x, let s say for instance we fin y = x 2 +3x then we can change the x variable an simply calculate the new value for y. y is epenent on x an we think of x, as being able to be change inepenently. For the same function y = x 2 +3x if we change y it is quite an involve process to fin the new value of x. Changing y results in a more involve calculation to fin x. Even though, strictly speaking, both variable are epenent on the other, through our equation; since we can t change one without the other, if we change the inepenent variable we can simply use our formula to fin the new value of the epenent variable. In some instances we may nee to fin the variable on the bottom of the erivative in terms of the variable on the top. In this case then the roles of the variables will be reverse. Here, to start with, we stick with the solution function on the top an the inepenent variable on the bottom of the erivative Alternative Notations for Differential Equations There are many ifferent notations for a erivative. We can write the erivative y x as y (x). So for a erivative of the form X (t) for instance, we want to fin the solution function X in terms of the inepenent variable t. For this alternative notation: For Y (x) we want to fin Y in terms of x. For X (t) we want to fin X in terms of t. For a ifferential equation of the form y x =3x2, we want to fin a solution function y as a function of x. For a ifferential equation of the form X (t) = 20X, we want to fin a solution function X as a function of t. Example 1 For the ifferential equation y x =4x2 +2x, ientify the solution function an the inepenent variable. Solution Here y is the solution function an x is the inepenent variable. We want to fin y as a function of x. Example 2 For the ifferential equation f (z) = 2z 2, ientify the solution function an the inepenent variable. Solution Here f is the solution function an z is the inepenent variable. We want to fin f as a function of z. Example 3 For the ifferential equation X (t) = 15X, ientify the solution function an the inepenent variable. Solution Here X is the solution function an t is the inepenent variable. We want to fin X as a function of t.

13 Draft Version August 2011 c 2010 University of Syney Exercises For the following ifferential equations fin the solution function(s) an the inepenent variable. Note: you o not nee to solve the ifferential equations here just name the solution function an the inepenent variable y x = x y x = y y x =3x2 +5y 2 y x = x x y = y2 +2x y x t x + t =0 x t =3x2 +4t 8. 0 = z y +2y z z x + [ ] 2 z =0 x y x +3y =0 11. x t + y = t, y t + x = t2 12. X (t) =4 13. X (t) =X(t) 14. X (t)+3x(t) =t X (Z)+Z X =0 16. Y (Z) + 3 = Y (Z) 17. X (t) Xt = X 3 t X (t) =X Y, Y (t) =Y X X t 2 + X t + X 20. X (t)+x (t)+x(t) =0

14 Draft Version August 2011 c 2010 University of Syney Differential Equations as Questions Differential Equations of the Form y x = f(x) as Questions With the solution function an inepenent variable in min we look at the rest of the ifferential equation. It is useful to think of a ifferential equation, not as a mathematical formula, but as asking a question in wors. Example 1 The ifferential equation is like asking the question: y x =2x Can you think of a function y, which is a function of x, such that when you ifferentiate that function you get 2x? Just one answer to this question is y = x 2, or y(x) =x 2. Since if you ifferentiate x 2 you get 2x. We coul have also chosen y(x) =x for instance. Example 2 The ifferential equation is asking: y x =3x2 Can you think of a function y, which is a function of x, such that when you ifferentiate that function you get 3x 2? Just one answer to that question is y = x 3. Can you think of another? Example 3 The ifferential equation is asking: y x =4x3 + 10x 3 Can you think of a function y, which is a function of x, such that when you ifferentiate that function you get 4x x 3?

15 Draft Version August 2011 c 2010 University of Syney Exercises For the following ifferential equations write own a simple sentence in wors that represents a question that the ifferential equation is asking y x =2 y x =2x y x = sin(x) y x =3x2 y x = ex y x = x sin(x) y +2x =3 x 8. X (t) =3t 2 9. Y (z) = sin z + z 10. W (t) + sin t = y x x =3 y sin(x) =x sin(x) x y x ex = e 2x Below are some harer questions that can be answere by extening the ieas in this section. [ ] 2 y = (3x 2 + 2) 2 x 15. X (t)+x(t) = 3t [X (t)] 2 + X (t) = ( ) 2 y y + sin(x) =0 x 18. Z (Y ) Z(Y )+3Z(Y )=Y y +3y + sin(x) =0 x 20. [X (t)] 2 + X(t) sin t = t 2

16 Draft Version August 2011 c 2010 University of Syney Differential Equations of the Form y x = g(y) as Questions In all of our examples so far the right han sie has only involve inepenent variables. For the example above there are only xs on the right. But there can be solution functions on the right as well. Example 1 For instance we can have a ifferential equation such as y x = y. This ifferential equation is, in essence, quite ifferent to the others we have iscusse above. The equation can be thought of as asking the question: Can you think of a function y, which is a function of x, such that when you ifferentiate that function you get the same function that you starte with? The function y = e x has erivative So y = e x is a function that has itself as its erivative. y x = x ex = e x = y. So y = e x is just one answer to our question what function is its own erivative. But again this is not the only function that answers our question. The function y =2e x has erivative So y =2e x is also a function that is its own erivative. So y =2e x is again just one answer to our question. y x = x 2ex =2e x = y. The functions y = e x an y =2e x are answers to our question an there are many more. Other answers to the question can you think of a function that is its own erivative are y =3e x, y = 24e x, y = 351e x. Inee the exponential function is unique as it is the only type of function, such that when you ifferentiate it you get back the same function. The only class of functions that are their own erivatives are the exponential functions of the form y(x) =Ae x. Since y x = x Aex = Ae x = y that is when we ifferentiate Ae x we get back the same function we starte with. Hence Ae x is the most general answer to our question. The exponential function is the only type of function who s erivative is the same as the function itself. y(x) = constant e x is the only type of function that is its own erivative.

17 Draft Version August 2011 c 2010 University of Syney 17 Example 2 The ifferential equation is more complicate again. It is asking: y x =4y Can you think of a function y, which is a function of x, such that when you ifferentiate that function, you get 4 times the function that you first thought of? Just one function which answers our question is y = e 4x, since here y x = x e4x =4 e 4x which is 4 times the function we first thought of. Can you think of others (Example 1 provies a hint)? Example 3 The ifferential equation is asking: x t = x2 +2x Can you think of a function x, which is a function of t, such that when you ifferentiate that function you get the square of the function you first thought of plus 2 times the function that you first thought of? The function which answers our question is x(t) = 2 ce 2t +1. As we can see the solutions to ifferential equations get very complicate very quickly. In fact we can write own very simple looking ifferential equations that o not have solutions in terms of simple functions. One such example is f (x) =e x2. There are many more. In fact writing own ifferential equation can be thought of as a way of efining many special functions. The ifferential equation y x = y can be use as a efinition for the function y(x) =ex an can be use to fin the important constant e = Differential Equations of the Form y x Example 4 = f(x)g(y) as Questions The ifferential equation is asking: x t =2tx Can you think of a function x, which is a function of t, such that when you ifferentiate that function you get 2t times the function that you first thought of? Just one function which answers our question is x = e t2, since here x t = e t2 =2t e t2 which is 2t times t the function we first thought of. Please note that you are not expecte to solve these ifferential equations here. You shoul unerstan how to unerstan what a ifferential equation may be asking.

18 Draft Version August 2011 c 2010 University of Syney Exercises 1. Which question is the following ifferential equation asking in wors? y x =3x2 (a) Can you think of a function such that when you ifferentiate that function you get 3 times the function. (b) Can you think of a function such that when you ifferentiate that function you get the square of the function. (c) Can you think of a function such that when you ifferentiate that function you get 2 times the square of the function. () Can you think of a function such that when you ifferentiate that function you get 3 times the square of the function. (e) None of the above. (f) All of the above. 2. Which question is the following ifferential equation asking in wors? y x =3y2 (a) Can you think of a function such that when you ifferentiate that function you get 2 times the square of the function you first thought of. (b) Can you think of a function such that when you ifferentiate that function you get 3 times the function you first thought of. (c) Can you think of a function such that when you ifferentiate that function you get 3 times the square of the function you first thought of. () Can you think of a function such that when you ifferentiate that function you get the square of the function you first thought of. (e) None of the above. 3. Which question is the following ifferential equation asking in wors? x t =3x2 (a) Can you think of a function such that when you ifferentiate that function you get 3t 2 (b) Can you think of a function such that when you ifferentiate that function you get 3 times the function you first thought of. (c) Can you think of a function such that when you ifferentiate that function you get the square of the function you first thought of. () Can you think of a function such that when you ifferentiate that function you get 3 times the square of the function you first thought of. (e) Can you think of a function such that when you ifferentiate that function you get 2 times the square of the function you first thought of. (f) None of the above. 4. Which question is the following ifferential equation asking in wors? x t =3x2 + x 3 (a) Can you think of a function such that when you ifferentiate that function you get 3t 2 + t 3 (b) Can you think of a function such that when you ifferentiate that function you get 3 times the function you first thought of plus that function cube.

19 Draft Version August 2011 c 2010 University of Syney 19 (c) Can you think of a function such that when you ifferentiate that function you get two times the square of the function you first thought of plus that function cube. () Can you think of a function such that when you ifferentiate that function you get 3 times the square of the function you first thought of plus that function square. (e) Can you think of a function such that when you ifferentiate that function you get 3 times the square of the function you first thought of. (f) None of the above. 5. Match up the ifferent ifferential equations with their corresponing questions, written in wors. (a) Can you think of a function such that when you ifferentiate that function you get 3t 2 + t 3 (b) Can you think of a function such that when you ifferentiate that function you get 2 times the function you first thought of plus that function cube. (c) Can you think of a function such that when you ifferentiate that function you get two times the square of the function you first thought of plus that function cube. () Can you think of a function such that when you ifferentiate that function you get 3 times the square of the function you first thought of plus that function square. (e) Can you think of a function such that when you ifferentiate that function you get 3 times the square of the function you first thought of. (i) (ii) (iii) (iv) (v) f t =2f 2 + f 3 x t =4x2 y t =3t2 + t 3 x t x t =3x2 =2x + x3

20 Draft Version August 2011 c 2010 University of Syney Particular Solutions an General Solutions or Differentiating in Reverse The ifference between General Solutions an a Particular Solution For this section we will concentrate on ifferential equations that look like y x = f(x), that is equations with the erivative on the left an a function of only the inepenent variable on the right. Examples of these type of ifferential equations are y x =3x2 +2x or y x = 10x4 sin(x), but with no solution functions, or ys here, on the right. Example 1 The ifferential equation y x that function you get 5. = 5 is asking; can you think of a function y(x) such that when you ifferentiate Just one answer is y(x) = 5x as here 5x =5 x If you know how to ifferentiate, answering these questions seems like we just nee to apply the rules of ifferentiation in reverse. This process is given the name anti-ifferentiation. Anti-ifferentiation simply means apply the rules of ifferentiation in reverse, in orer to solve our ifferential equations or answer our questions. Seems simple so far, though there can be many answers to one question, just like the meaning of life. Example 2 The ifferential equation y =2x is asking; what y o we ifferentiate to get 2x? x We know y(x) =x 2 is an answer to this question as x x 2 =2x. So we have one solution to this ifferential equation. But if we ifferentiate y = x for instance, we also have x (x2 + 1) = 2x. This means that y = x is also a solution to our ifferential equation. But so is y = x 2 1, y = x , y = x , y = x , y = x 2 + π an y = x 2 π There can be many solutions to just one ifferential equation. If we ifferentiate any constant we get 0. So if we ifferentiate we get y = x 2 + any constant x (x2 + any constant) = 2x. So there are lots of answers to our question y =2x. In fact there are an infinite number of them. x The most general solution to this ifferential equation is y = x 2 + c, where c is a constant.

21 Draft Version August 2011 c 2010 University of Syney 21 The constant c, as use here, is calle the constant of integration. Since there may be lots of solutions to any one ifferential equation we use special names for each type. The solution y = x 2 + c is calle the general solution of the ifferential equation y x =2x. As it is the most general solution to the ifferential equation. Any one of these solutions on there own, such as, y = x 2, y = x , y = x 2 π are all still solutions to the ifferential equation. Any one of these solutions is calle a particular solution to the ifferential equation. y = x 2 is a particular solution to the ifferential equation. y = x is also a particular solution to the ifferential equation. y = x 2 π is also a particular solution to the ifferential equation. The solution y = x 2 is calle a particular solution of the ifferential equation y x =2x. As it is the just one solution to the ifferential equation Exercises 1. For the ifferential equation which of the below are particular solutions? (a) y = x 3 (b) y = x (c) y = x 3 π () y = x (e) y = x 3 c (f) None of the above. (g) All of the above. 2. For the ifferential equation which of the below are particular solutions? (a) y = x (b) y = x 4 + π (c) y = x 3 + c () y = x 3 c (e) y =4x 3 + c (f) None of the above. (g) All of the above. 3. For the ifferential equation which of the below are general solutions? (a) X(t) =cos(t)+c (b) X(t) = cos(t) + 10 (c) X(t) = cos(t)+c () X(t) =cos(t)+2 (e) X(t) = cos(t)+2 (f) None of the above. (g) All of the above. y x =3x2 y x =4x3 X (t) = sin(t)

22 Draft Version August 2011 c 2010 University of Syney Particular Solutions an General Solutions in General The most general solution to a ifferential equation is calle the general solution. For the ifferential equations with only the first erivative, in these notes, the general solution will involve a constant such as c above. The constant c, as use here, is calle the constant of integration. Any single solution to a ifferential equation is calle a particular solution. The particular solution will not involve a constant of integration, such as c above. Summary of Terminology For Differential Equations y = f(x) is calle a ifferential equation. There are many other types. x y = f(x) x is calle the general solution of the ifferential equation. The integral will involve a constant c say. c is calle the constant of integration. If we evaluate c, for instance if c = 0 then y = f(x) x, with c = 0, is calle a particular solution of the ifferential equation. The process of answering a question about the erivative of a function that we are using here is calle anti-ifferentiation. Example y x =5x2 is calle a ifferential equation. y = 5 3 x3 + c is calle the general solution of the ifferential equation. c is calle the constant of integration. If we evaluate c, for instance if c = 0 then y = 5 3 x3 is calle a particular solution of the ifferential equation. y = 5 3 x is also a particular solution. The process we are using of answering a question about the erivative of a function is calle anti-ifferentiation.

23 Draft Version August 2011 c 2010 University of Syney Exercises The following mathematical expressions, that are labelle (a) to (f), are just part of a solution to a problem involving a ifferential equation. Match up each mathematical expression with the appropriate escription in wors, labelle (i) to (vi). Please note: In general you shoul provie answers with escriptions of the solution in wors as well as mathematics. As an example of how to set out solutions in wors an mathematics see Ten Important Steps for Solving Moeling Questions Pose in Wors 1. For the ifferential equation y x =2x (a) y x =2x (b) y = x 2 + c (c) y = x 2 +5 () c (e) c =5 (f) y = x 2 2. For the ifferential equation y x =5x4 (a) c =5 (b) y = x 5 + c (c) y = x 5 +5 () c (e) y x =5x4 (f) y = x 5 + c (i) Differential Equation (ii) Particular Solution (iii) General solution (iv) Particular Solution (v) Constant of Integration (vi) Evaluation of constant of integration (i) General solution (ii) Particular Solution (iii) Differential Equation (iv) Constant of Integration (v) General Solution (vi) Evaluation of constant of integration 3. For the escription of a ifferential equation, given by the following escription; Can you think of a function such that when you ifferentiate that function with respect to t you get sin(t). match up each mathematical expression labelle (a) to (f) with the appropriate escription in wors, labelle (i) to (vi), if an appropriate escription exists. (a) X = cos(t)+c (b) X = cos(t) (c) t x = sin(t) () X = cos(t)+c (e) X t = sin(t) (f) X = cos(t) + 15 (i) General solution (ii) Particular Solution (iii) Differential Equation (iv) Constant of Integration (v) Particular Solution (vi) Evaluation of constant of integration

24 Draft Version August 2011 c 2010 University of Syney For the escription of a ifferential equation, given by the following escription; Can you think of a function such that when you ifferentiate that function with respect to x you get xe x + e x. match up each mathematical expression labelle (a) to (f) with the appropriate escription in wors, labelle (i) to (vi), if an appropriate escription exists. (a) y x = xex + e x + c (b) Y = xe x + c (c) Y = xe x 42 () Y (x) =xe x + e x + c (e) Y (x) =xe x + e x (f) Y (x) =xe x + e π (i) General solution (ii) Particular Solution (iii) Differential Equation (iv) Constant of Integration (v) Particular Solution (vi) Evaluation of constant of integration

25 Chapter 3 Moelling an Calculus 3 MAC 3 Solving some ifferential equations by using the concepts an interpretations of a ifferential equation to fin a solution or an answer pose by the ifferential equation. = f(x) Using Dif- 3.1 Solving Differential Equations of the Form y x ferentiation in Reverse The most important step in solving ifferential equations, at least at the beginning, is to unerstan what a ifferential equation represents an what it is asking. The secon most important step in starting to solve ifferential equations is unerstaning the terminology an notation use. We now use these ieas to solve some simple ifferential equations. Here we concentrate on ifferential equations of the form y x = f(x). For these ifferential equations the right han sie is always some function of x, like y x To solve a ifferential equation of this form we are aske the question: What function y, which is a function of x, has a erivative given by f(x). =2x, y x =3x2 etc. If you know various erivatives this may be just a matter of, remembering erivatives, or looking up ifferentiation tables. Example 1 Fin one solution of the ifferential equation y x = cos(x). From the ifferentiation tables sin(x) = cos(x), so a solution is y(x) = sin(x). x 25

26 Draft Version August 2011 c 2010 University of Syney 26 Example 2 For the ifferential equation fin one particular solution. F (t) =3e 3t From the ifferentiation tables t e3t =3e 3t, so a particular solution is F (t) =e 3t.

27 Draft Version August 2011 c 2010 University of Syney Exercises By using the ifferentiation tables ientify which one of the options are particular solution to the ifferential equations (a) y = 10x + c (b) y = 10x 2 (c) y = 10x c () y = 10x π (e) None of the above. (a) y =6x 2 (b) y =6 x 2 /2+c (c) y =3x 2 + c () y =3x 2 (e) None of the above. (a) y =3x 5 (b) y =3 x 5 /5 (c) y =3 x 5 /5+c () None of the above. (a) y =6x 6 (b) y =6x 6 + c (c) y =6 x 6 /5 () y = x 6 + c (e) None of the above. y x = 10 y x =6x y =3 5x4 x y x =6x5 5. (a) y = 10x 14 + cπ (b) y = 10x 14 + c (c) y = 10x π () y = 10x 14 (e) all of the above (f) none of the above. y = 10 14x13 x

28 Draft Version August 2011 c 2010 University of Syney X t = et (a) y = e x (b) y = e x + c (c) y = e t + c () X = e t + π (e) none of the above. 7. Y (x) =5x 4 +3x 2 (a) Y (x) =x 5 + x 3 (b) Y (x) =x 5 + x (c) Y (x) =x 5 + x 3 e 42π () Y (x) =x 5 + x 3 + c (e) none of the above. 8. X (t) = 1 t (a) X (t) = ln(t) (b) X(t) = ln(t) + c (c) X(t) = ln(t) e 355π+ln(42) () Y (x) = ln(x) 1 (e) none of the above. 9. y x = sec2 (x) (a) y = sec(x) (b) y = sec 2 (x) (c) y = tan(x)+c () y(x) = tan(x) 53 (e) none of the above. 10. X t =2t sec2 (t 2 ) (a) X(t) = tan(t 2 ) (b) X = tan(t 2 ) 51π (c) X = tan(t 2 ) () X = tan(t 2 )+0 (e) none of the above.

29 Draft Version August 2011 c 2010 University of Syney Differentiation Tables No. General Rule Example 1 Example 2 Example 3 (1) (of any constant c) = 0 x x 1 = 0 x 10 = 0 x π = 0 (2) x x =1 t t =1 z z =1 (3) x x2 =2x t t2 =2t f f 2 =2f (4) x x3 =3x 2 t t3 =3t 2 u u3 =3u 2 (5) x xn = nx n 1 x x4 =4x 3 x x 1 = 1x 2 = 1 x 2 x x16 = 16x 15 (6) f (x) cf(x) =c x x 10x = 10 1 = 10 x x 2x2 =2 2x =4x x πx0.3 = π 0.3x 0.7 (7) x ex = e x t et = e t u eu = e u z π ez = π e z (8) x eax = ae ax t e3t =3e 3t u e10u = 10e 10u z e 2.3z = 2.3e 2.3z (9) x ef(x) = e f(x) x f(x) =f (x)e f(x) 2 x ex =2xe x2 =6x 2 e x e2x3 2x3 = 6πu 7 e u eπu 6 πu 6 (10) x ln(x) = 1 x t 3 ln(t) =3 1 t w π ln(w) = π w (11) x ln(f(x)) = 1 f(x) x f(x) (x) =f f(x) x ln(x2 )= 1 x 2 2x x ln(5x4 )= 1 5x 4 20x3 x 3 ln(πx 6 )= 3 6πx 7 πx 6

30 Draft Version August 2011 c 2010 University of Syney 30 Differentiation Tables Continue No. General Rule Examples (12) (13) sin(x) = cos(x) x x sin( f(x) ) = cos( f(x)) x f(x) = cos( f(x))f (x) sin(t) = cos(t) t 355 sin(x) = 355 cos(x) x z π2 sin(z) = π 2 cos(z) sin(2x) = cos(2x) 2 x (14) cos(x) = sin(x) x x sin(5x2 4x) = cos(5x 2 4x) (10x 4) x 24 sin(e3x ) = 24 cos(e 3x ) 3e 3x 24 cos(x) = 24 sin(x) x z ( 3π5 cos(z)) = 3π 5 sin(z) f πe5 cos(f) = πe 5 sin(f) (15) x cos( f(x) ) = sin( f(x)) x f(x) = sin( f(x))f (x) cos(2x) = sin(2x) 2 x x cos( 4x3 +3x 2 5x) = sin( 4x 3 +3x 2 5x) ( 12x 2 +6x 5) x 24 cos(e πx )= 24 sin(e πx ) πe πx = 24πe πx sin(e πx )

31 Draft Version August 2011 c 2010 University of Syney 31 Differentiation Tables Continue No. General Rule Examples (16) (17) x tan(x) = sec2 (x) x tan( f(x) ) = sec2 ( f(x)) x f(x) = sec 2 ( f(x))f (x) x tan(t) = sec2 (t) x 42 tan(x) = 42 sec2 (x) z 4π2 tan(z) = 4π 2 sec 2 (z) x tan(x3 ) = sec 2 (x 3 ) 3x 2 x tan(4x3 3x + 2) = sec 2 (4x 3 3x + 2) (12x 2 3) x 5692π2 tan(e 53πx )= 5692π 2 sec 2 (e 53πx ) 53πe 53πx

32 Draft Version August 2011 c 2010 University of Syney Solving Differential Equations of the Form y = f(x) Using Integration x A ifferential equation of the form y = f(x) may be solve by using ifferentiation in reverse that is by x anti-ifferentiation by asking: What function y(x) can be ifferentiate to give f(x)? But we can also try to solve this type of equation by using integration. We can uno the on the left by integrating. If the function on the right, that is f(x) can be integrate x (over the interval of x) then we can fin a solution to the ifferential equation. We start with the ifferential equation y x = f(x). We then integrate both sies of the equation with respect to the inepenent variable, here x. Given certain conitions if we performing the same operation to both sies of the equation then the two sies of the equation will remain equal. This is analogous to multiplying both sies of an equation by 2, or taking the square of both sies of the equation. So long as we perform precisely the same operation to both sies of the equation the equality remains vali. You must always integrate both sies of the equation with respect to the same variable. Giving the solution; y x x = f(x) x. y = f(x) x. For this special case then we can use integration or integration tables to fin a solution for the ifferential equation. Example 1 Question Fin the general solution to the ifferential equation Answer y x = x2 Step 1 Integrate both sies of the ifferential equation with respect to the inepenent variable x, y x x = x 2 x Step 2 perform the integration, where c is a constant. y = Step 3 State the answer in wors an mathematics. The general solution is x 2 x = 1 3 x3 + c, y = 1 3 x3 + c.

33 Draft Version August 2011 c 2010 University of Syney 33 Example 2 Question Fin the general solution to the ifferential equation Answer X t = 2t 3 t 2 Step 1 Integrate both sies of the ifferential equation with respect to the inepenent variable t, X t t = 2t 3 t 2 t Step 2 perform the integration, where c is a constant. 2t X(t) = 3 t 2 t = ln 3 t 2 + c, Note: here we are using the rule that says if we have a function on the bottom of a fraction an the erivative of that function on the top then the integral of that fraction is the natural log of the absolute value of that function or in the language of mathematics : f (x) x = ln( f(x) )+c f(x) So for our integral we nee the 2t on the top to use this rule. Step 3 State the answer in wors an mathematics. The general solution is where c is a constant. X(t) = ln 3 t 2 + c, In General: To solve a ifferential equation of the form y = F (x) we may be able to fin the integral x of F (x). Lets say F (x) x = f(x)+c. The steps are: Step 1 Integrate both sies of the ifferential equation y x x = F (x) x Step 2 perform the integration, y = F (x) x = f(x)+c, where c is a constant. Step 3 State the answer in wors an mathematics. The general solution is y = f(x)+c.

34 Draft Version August 2011 c 2010 University of Syney Exercises By integrating both sies of the ifferential equations below, fin the most general solutions. The integration tables may be useful for fining the integrals. Note for each of these exercises you shoul set out the solution inicating in wors what you are oing at each stage. The explanation of what you are oing in wors is often aware marks in an examination y x =3x3 y x =2 y x =5x6 16x 3 y x = πx2 y x = cos(x) y x = e5x y =2x + cos(x) x y x = cos(3x) y x = cos(x2 ) 2x y x = 2 ex 2x y x = 1 x 5 5x4 + sin(3x 5 ) 15x 4