Paper II ( CALCULUS ) Prof. R. B. Patel Dr. B. R. Ahirrao Prof. S. M. Patil Prof. A. S. Patil Prof. G. S. Patil Prof. A. D. Borse Art, Science & Comm. College, Shahada Jaihind College, Dhule Art, Science & Comm. College, Muktainagar Art, Science & Comm. College, Navapur Art, Science & Comm. College, Shahada Jijamata College, Nandurbar
Unit I Limit, Continuity, Differentiability and Mean Value Theorem Q. Objective Questions Marks. 5 lim 5 + 5 is equal to a) b) c) d) none of these. cos lim is equal to a) b) c) - d) none of these tan. Evaluate lim a) b) c) d). The value of the log(sin ) lim log(sin ) is a) b) c) d) - 5. lim e n is equal to a) b) - c) d) 6. log(sin a) lim,( ab, > ) is equal to log(sin b) a) - b) c) d) none of these 7. lim log is equal to a) b) c) d) - 8. lim sin is equal to a) b) c) - d) none of these
9. lim e is equal to a) b) c) d). lim is equal to a) b) - c) d) none of these. lim ( tan ) tan is a) e b) e c) e d) e. The function f ( ) = sin, for and f() =, for = a) Continuous and derivable b) Not continuous but derivable c) Continuous but not derivable d) Neither continuous nor derivable at the point =. The function f ( ) = sin, for and f() =, for = is a) Continuous and derivable b) Not continuous but derivable c) Continuous but not derivable d) Neither continuous nor derivable. For which value of c ( a, b), the Roll s theorem is verified for the function + ab f ( ) = log defined on a, b a ( + b) [ ] a) Arithmetic mean of a & b b) Geometric mean of a & b c) Harmonic mean of a & b d) None of these.
5. For which value of c ( a, b) = (, ), the Rolle s theorem is applicable for the function f( ) = sin, in[, ] a) b) c) d) 6. For which value of c,, the Rolle s theorem is applicable for the function f( ) = sin+ cos in, a) b) c) d) 6 7. For which value of c (, 5), the Rolle s theorem is verified for the function f in [ ] ( ) = 6 + 5,5 a) b) c) d) 8. for which value of c (-, ). the L.M.V.T. is verified for the function f in [ ] ( ) = +, a) b) c) d) 9. L.M.V.T is verified for the function f( ) = 7+ in[,5] a) 5 b) c) d) 7. For which value of c, C.M.V.T. is applicable for the function f() = sin, g() = cos in [, /] a) b) c) 6 d). If the C.M.V.T. is applicable for the function f() = e, g() = e -, in [a, b] find the value of c ( a, b) a) a+ b b) ab c) a + b d) none of these
. If the C.M.V.T. is applicable for the function f() =/, g() = /, in [a, b] find the value of C. a) a+ b b) ab c) ab a+ b d) none of these. If log log 5 f( ) =, 5 5 is continuous at = 5 then find f(5) a) 5 b) -5 c) 5 d) 5 sin. If f( ) =, ( ) is continuous at = then f(/) is a) 8 b) c) d) - 5. If cos f( ) =, is continuous at = then value of f() is sin a) b) c) - d) none of these a a a 6. I f f ( ) =, a a is continuous at = a, then find f(a) a a a) a log a b) a log a c) log a d) none of these 7. Evaluate lim sin log a) b) c) - d) 8. Evaluate lim tan log. a) b) c)- d) none of these 9. lim log is equal to a) b) c) d)-. lim cos is equal to a) - b) c) d)
. Evaluate lim a) b) c) d) -. Evaluate lim a). Evaluate b) lim( + ) c) d) - a) - b) c) - d). If sin ( ) f( ) =, f() is continuous at point = find a) b) c) d) none of these 5. Evaluate sin e e lim sin a) b) - c) d) -
Q. Eamples Marks. Evaluate. Evaluate. Evaluate. Evaluate 5. Evaluate 6. Evaluate tan lim sin e + lim log( ) lim log(tan ) log(tan ) lim cot lim a a tan( a ) lim(cot ), > 7. Evaluate 8. Evaluate lim(cot ) log lim tan 9. Eamine for continuity, the function f ( ) = a, a for < < a =, for = a = a, for> a. Using δ definition, prove that f( ) = cos, if =, if = is continuous at =. Eamine the continuity of the function e f( ) =, e + if =, if = at the point =.
. Eamine the continuity of the function 9 f( ) =, for < = 6, for = 8 = 8, for > at the point =.. Eamine the continuity of the function. If the function f( ) =, for < < =, for = 6 =, for > at the point =.. sin f( ) = + a, for > 5 = + b, for < =, for = is continuous at =, then find the values of a & b. 5. If f() is continuous on [, ] f( ) = sin, for - = αsin + β, for - < < = cos, for Find α & β. 6. Define differentiability of a function at a point and show that f ( ) continuous, but not derivable at the point =. 7. Discuss the applicability of Rolle s Theorem for the function [ ] = is ( ) ( ) m n f = a ( b) defind in a, b where m, n are positive integers.
8. Discuss the applicability o Rolle s Theorem for the function 5 f( ) = e (sin cos ) in,. 9. Verify Langrange s Mean Value theorem for the function f( ) = ( )( )( ) defined in the interval[, ].. Find θ that appears in the conclusion of Langrange s Mean Value theorem for the function f( ) =, a =, h=.. Show that b a b a < tan b tan a<, if < a< b. + b + a And hence deduce that + < tan < + 5 6 a b b. For < a < b, Prove that < log < and hence show that b a a 6 < log < 6 5 5. If < a < b <, then prove that b a b a < sin b sin a< a b Hence show that < sin < 6 6 5. Show that + < < > tan, 5. For >, prove that < log( + ) < ( + ) 6. Separate the interval in which f ( ) = + 8 + 5 is increasing or decreasing. 7. Show that log( ), + < + < 8. Show that tan < <, > + 9. With the help of Langrange s formula Prove that α β tan tan α < α β < β, where β α cos β cos α
. Verify Cauchy s Mean Value theorem for the function f () = sin, g () = cos in. Show that sin α sin β = cot θ, where < α < θ < β < cos β cosα. If f ( ) = and g( ) = in Cauchy s Mean Value Theorem, Show that C is the harmonic mean between a & b.. Discuss applicability of Cauchy s Mean Value Theorem for the function f () = sin and g () = cos in [ ab, ].. Verify Cauchy s Mean value theorem f ( ) =, g( ) = in[ a, b] 5. Find c (,9) such that f (9) f() f '( C) = g(9) g() g'( c) where f ( ) = and g( ) = 6. Discuss the applicability o Rolle s Theorem for the function + f( ) = log in, ( ). 7. Verify Langrange s Mean Value theorem for the function f( ) = ( )( ) in, 8. Discuss the applicability o Rolle s Theorem for the function - f( ) = e cos in,. 9. Verify Langrange s Mean Value theorem for the function [ ] f( ) = + 9 in,7.
Q. Theory Questions Marks / 6. If a function f is continuous on a closed and bound interval [ a, b],then show that f is bounded on [a, b].. Show that every continuous function on closed and bounded interval attains its bounds.. Let f :[ ab, ] R be a continuous on [a,b] and if f ( a) < k < f( b), then show that there eists a point c ( a, b) such that f () = k.. If f () is continuous in [a, b] and f ( a) f( b), then show that f assume every value between f (a) and f (b). 5. If a function is differentiable at a point then show that it is continuous at that point. Is converse true? Justify your answer. 6. State and Prove Rolle s theorem OR If a function f() defined on [a,b] is i)continuous on [a,b] ii) Differentiable in (a, b) iii) f (a ) = f( b) then show that there eists at least one real number c ( a, b) such that f (c)=. 7. State and Prove Langrange s Mean Value Theorem. OR If a function f() defined on [a,b] is i) continuous on [a,b] ii) differentiable in (a, b) then show that there eit at least one real number c ( a, b) such that f '( c) = f ( b) f( a) b a 8. State and Prove Cauchy s Mean Value Theorem. OR If f() and g() are two function defined on [a,b] such that i) f() and g() are continuous on [ a, b] ii) f() and g() are differentiable in (a,b) iii) g '( ), ( a, b) then show that there eist at least one real number c ( a, b) such that f '( c) f( b) f( a) = g '( c) g( b) g( a)
9. State Rolle s Theorem and write its geometrical interpretation.. State Langrange s Mean Value Theorem and write its geometrical interpretation.. If f() is continuous in [a,b] with M and m as its bounds then show that f() assumes every value between M and m.. Using Langrange s Mean Value Theorem show that cos aθ cosbθ b a, ifθ θ. If f() be a function uch that f '( ) =, ( a, b) then show that f() is a constant in this interval.. If f() is continuous in the interval [a,b] and f '( ) >, ( a, b) then show that f() is monotonic increasing function of in the interval [a,b]. 5. If a function f() is such that i) it is continuous in [a, a+h] ii) it is derivable in (a, a+h) iii) f(a) = f(a+h) then show that there eist at least one real numberθ such that f '( a+ θ h) =, where <θ <. 6. If the function f() is such that i) it is continuous in [a, a+h] ii) it is derivable in (a, a+h) then show that there eists at least one real number θ such that f( a+ h) = f( a) + hf '( a+ θ h), where< θ < 7. If f() is continuous in the interval [a,b] and f '( ) <, ( a, b) then show that F() is monotonic decreasing function of in the interval [a, b].
Unit II Successive Diff. And Taylor s Theorem, Asymptotes, Curvature and Tracing of Curves Q-.Question (-marks each). State Leibnitz theorem for the. Write th n derivative of e a. th n derivative of product of two functions.. Write. Write th n derivative of sin( a + b). th n derivative of cos( a + b). 5. State Taylor s theorem with Langrange s form of reminder after th n term. 6. State Maclaurin s infinite series for the epansion of f() as power series in [,]. 7. Define Asymptote of the curve. 8. Define intrinsic equation of a curve. 9. Define point of infleion.. Define multiple point of the curve.. Define Double point of the curve.. Define Conjugate point of the curve.. Define Curvature point of the curve at the point.
Q- Eamples ( - marks each). If. If. If. If + + y =, find y. n + y = e find y n a cos sin,. y = + find y8 sin( 7),. y = (sin ) Provethat ( ) y (n+ ) y n y = n+ n+ n 5. If y = cos( msin ) P rove that ( ) y (n+ ) y + ( m n ) y = n+ n+ n If y = tan(log y) P rove that 6. ( + ) y + (n ) y + n( n ) y = n+ n n 7. m m If y +y = P rove that ( ) y + (n+ ) y + ( n m ) y = n+ n+ n cos y log P rove that If ( ) = ( ) n 8. b n y + (n+ ) y + ny = n+ n+ n 9. Find. Find yn if y = ( + )(+ ) yn if y = cos If y = a cos(log ) + bsin(log ) P rove that. y + (n+ ) y + ( n+ ) y = n+ n+ n. If y = tan P rove that ( + ) yn+ + ( n+ ) yn+ + n( n+ ) yn =
. Find y if y = e log n. Find y if y = cos cos cos n 5. If y = sin cos P rove that ( n ) n yn =.cos + 8 6. If n y=( -) P rove that ( ) yn+ + yn+ n( n+ ) yn = 7. mcos If y = e P rove that ( ) y (n+ ) y ( n + m ) y = n+ n+ n = 8. ( + ) m If y a P rove that ( a ) y + (n+ ) y + ( n m ) y = n+ n+ n 9. If y = sin( msin ) P rove that ( ) y (n+ ) y ( n m ) y = n+ n+ n If y = cos(log ) P rove that. y + (n+ ) y + ( n+ ) y = n+ n+ n. Use Taylor s theorem to epress the polynomial of ( - ).. Epand sin in ascending powers of ( ) 7 6 + + in powers. Assuming the validity of epansion, prove that 5 e cos = + + 6
. Assuming the validity of epansion, prove that 5 6 sec = + + +!! 6! 5. Epand log(sin) in ascending powers of ( - ). 6. Epand tan in ascending powers of ( ) 7. Prove that 8. Prove that tan = + -------- and hence find the value of 5 5 5 7 sin = +. +.. +..5. =! 5! 7! 9. Use Taylor s theorem,evaluate ( ) tan sin lim 5. Epand. Epand e in ascending powers of ( - ). 7 5 6 + + in power of ( - ).. Obtain by Maclurin s theorem the first five term in the epansion of log( + sin ).. Obtain by Maclurin s theorem the epansion of log( + sin ) upto.. Assuming the validity of epansion, prove that sin e = + + + 8 5. Find the asymptotes of the curve y = 6. Find the asymptotes of the curve y = + + 9 7. Find the asymptotes of the curve 8. Find the asymptotes of the curve y = = t, y = t+ tan t 9. Find the asymptotes of the curve. Find the asymptotes of the curve y = y = +
. Find the differential arc and also the cosine and sine of the angle made by the tangent with positive direction of X-ais. for the curve. Find the differential of the arc of the curve r acos ( ) θ y = a. =.Also find the sine ratio of the angle between the radius vector and the tangent line.. Find the point on the parabola y = 8 at which curvature is.8.. Find the curvature of r = a cos θ, at θ =. 5. Find the curvature and radius of curvature at a point t on the curve, = a(cost+ tsin t), y = a(sin t tcos t). 6. Find the curvature of the curve, y = at P(, ). 7. Find the curvature of the curve, y at = 8 origin. 8. Find the curvature of the curve, y = at P(,). 9. Eamine for concavity and point of inflection of Guassian Curve y = 5. Trace the curve 5. Trace the curve y = + ( ) ( ) y = ( ) 5. Find the asymptotes parallel to co-ordinate aes for the curve y ( a ) = 5. Find the radius of curve of y = ctanψ. 5. Show that the curvature of the point (, ) 8 + y = ay is a a a on the folium 55. Find the point on the parabola y = 8 at which radius of curvature is 6 7. 56. Eamine the nature of the origin of y ay + =. 57. Trace the curve + y = ay. 58. Trace the curve y = a ( a ).
Unit III Integration of Irrational Algebraic and Transcendental Functions, Applications of Integration Q- Marks -. The proper substitution for the integral of the type d ( p + q) a + b is. Evaluate d. Evaluate. Evaluate 5. Evaluate 6. Evaluate d ( ) + d ( ) d + d ( ) 7. Evaluate d ( ) 8. Evaluate 9. Evaluate. Evaluate d (+ ) cos d. (sin ) sin ed (e + ) e. Reduction formula for sin n d is
. Evaluate. Evaluate 9 sin d = ------------- sin 6 d. Evaluate sin 7 d 5. Reduction formula fpr cos n d = 6. Evaluate cos 8 d 7. Evaluate cos 9 d 8. Evaluate sin d 9. Evaluate sin 5 d. Evaluate a a 5 d d. Evaluate ( ) a + d. Evaluate 5 ( + ). Evaluate. Evaluate 5. Evaluate sin sin sin.cos.cos 6 5.cos 6 d d d 6. Evaluate sin.cos 5 7 d
7. Evaluate 8. Evaluate 9. Evaluate. Evaluate 5 sin.cos d 8 5 sin.cos d 8 sin.cos d 5 9 sin.cos d. The proper substitution for the integral of the type d ( p + q+ r) a + b+ c. The length of the arc of the curve y = f() between the points = a, = b is given by S = ---------------- with usual notation.. The length s of the arc of the curve = f (t),y = ψ (t) between the points where t = a, t = b is given by S = ------------ with usual notations.. The equation of the Catenary is ---------- 5. The equation of the Astriod is ---------- 6. The volume of the solid generated by revolving about X-ais, the area bounded by the curve y = f(), the X- ais and the ordinate = a, = b is given by V = ------------- with usual notation. 7. The volume of the solid generated by Revolving about X-ais,the area bounded by the curve = g (y), the Y-ais and the abscissas y = c, y = d is given by V= ---------------with usual notation. 8. The volume of the solid generated by revolving about X-ais, the area bounded by the parametric curve X = φ (t),y = ψ (t ) and the ordinate t = a, t = b is given by V = ------------- with usual notation. 9. The volume of the solid generated by revolving about Y-ais, the area bounded by the parametric curve Y= φ (t),y = ψ (t ) and the abscissas t = a, t = b is given by V = ------------- with usual notation.
. The volume of the solid generated by revolving about X-ais, the area bounded by the curve Y = φ ( ),Y = ψ ( ) and the ordinates = a, = b is given by V = ------------- with usual notation.. The Volume of the sphere of radius a is --------------. The volume of the ellipsoid formed by revolving the ellipse about Y-ais is -------- a y b + =. The area of the curved surface of the solid generated by revolving about X- ais, the area bounded by the continuous curve y = f( ), the X-ais and the ordinates = a, = b is S=-------------. The area of the curved surface of the solid generated by by revolving about Y-ais, the area bounded by the continuous curve g = f( y), the Y-ais and the abscissae y = c, y = d is S=------------- 5. The area of the curved surface of the solid generated by by revolving about X-ais, the area bounded by the curve = φ (t), y = ψ ( t), the X-ais and the ordinates t = a, t = b is S =------------- where ds dt = 6. The surface area of the sphere of radius a is ----------- 7. Write down the parametric equation of the cycloid. 8. 9. n d = ( ) d = + n 5. Define i) A rectification ii) A cap of the sphere.
Q- (-marks each) Integral of the form d p + q+ r a+ b ( ). Evaluate d ( + ). Evaluate. Evaluate. Evaluate 5. Evaluate 6. Evaluate d + ( ) d ( + ) d ( + 5+ 8) + d ( + ) d ( + 5) Integral of the form ( ) d p + q a + b + c 7. Evaluate 8. Evaluate 9. Evaluate. Evaluate. Evaluate. Evaluate d + + d ( ) + d + + d ( ) + d ( ) + d ( + ) +. Evaluate d ( ) + +, ( )
. Evaluate d 5. Evaluate d ( + ) + + Integral of the form d ( ) p + q + r a + b + c 6. Evaluate 7. Evaluate 8. Evaluate 9. Evaluate d ( + ) d ( + ) + d ( ) + d ( + ) + Reduction formula type eamples-. Evaluate. Evaluate. Evaluate d ( + ) 9 7 ( ) d a a d. Evaluate d d. Evaluate 5 ( + )
5. Evaluate 8 d 6. Evaluate 7. Evaluate 8. Evaluate 9. Evaluate. Evaluate. Evaluate. Evaluate. Evaluate. Evaluate 5. Evaluate 6. Evaluate d 6 7 + d d 7 ( + ) 7 ( + ) 7 ( + ) 5 9 ( + ) ( + ) ( + ) + + d d d d d d 6 d 5 d 7. Show that 8. show that d = d 5 = 8
9. show that 6 6 d 7 =. show that d 7 = 8. Let I n =, show that I n = + I n- Where n is a positive integer.. sin 6 sin 5 sin Show that d = sin sin + + 5 sin 6 Hence S how that d = sin. Show that sin 7 sin 6 sin sin d = sin + + + 6 sin 7 Hence S how that d = sin. Let I = sin d, sin Show that I sin sin9 = + + I 9 8 5. sin 5 sin Show that d sin ( sin ) = +
Q. (6-marks each) Reduction formulae m. Evaluate sin.cos. Evaluate. Evaluate. Evaluate n d, where m, n are positive integers. sin n d, where n is positive integers. cos n d, where n is positive integers. m n (sin ).(cos ) d, where m and n are positive integers. 5. Evaluate d, where n is a positive integers. ( ) n + + Application of Integration. Rectification 6. Show that the length of an arc of the parabola y = a cutoff by the y = is + log(+ ). 7. Show that the length of an arc of the parabola = y form the verte to any etremity of the latus rectum is log( ). + + 8. Show that the length of the arc of the curve y = cutoff by the line y = is 5 log( 5) + +. c c c 9. Find the length of an arc of the catenary y ( e e ) verte (, c) to any point (, y ). = + measured from the
. Find the length of an arc of the curve y = sin e between the points where y = and y =. 6. Using theory of integration, obtains the circumference of the circle + y = 5.. Find the length of an arc of the cycloid a( sin ), y e θ θ θ = θ θ = cos sin between the cups θ = and θ =.. Find the length of an arc of the curve e θ θ θ = sin + cos, y e θ θ θ = cos sin between the cups θ = and θ =.. Find the length of an arc of the curve = a(cosθ cos θ ), y = a(sinθ si θ ), measured from the points, where θ = and θ = is 89. 5. Find the length of an arc of the curve = a(cosθ + θsin θ), y = a(sinθ θ cos θ), from the points, where Volumes of Solids of Revolution θ θ = and = is a. 6. Using theory of integration, show that the volume of sphere of radius a is a cubic units. 7. Show that the volume of solid genered by revolving the ellipse a about X-ais is ab cubic units. y + =, b 8. Find the Volume of the solid formed by revolving the arch of the cycloid = a( θ sin θ ), y = a( cos θ ) about its base. 9. The area enclosed by the hyperbola y = and the line + y =7 is revolved about X-ais, Show that the volume of the solid generated is cubic units. Compute the volume of the solid generated by revolving about Y-ais, the region enclosed by the parabolas y and y = 8 =.
Areas of surface s of revolution-. The are of the parabola y = between the origin and the point (,) is revolved about X-ais, Find the area of the surface of revolution of the solid formed.. Find the surface area of the solid generated by the revolution about the X-ais t of the loop of the curve = t, y = t.. The arc of the parabola y = between its verte and an etremity of its latus rectum revolves about its ais. Find the surface area traced out.. If the segment of a straight line y = between = to = is revolved about Y-ais.show that surface area of the solid so formed is 5 square units. 5. Find the area of the surface generated when the segment of the straight line y = between = to = is revolved about Y-ais.
Unit IV Differential Equation of First Order & First Degree Q- or 6 Marks. Eplain the method of solving homogeneous diff. Equation of the type Md + Ndy =, where M = M(, y), N = N(, y). Eplain the method of solving non-homogeneous diff. Equation =, where a, b, c, a, b, c are real numbers.. Eplain the method of solving eact diff. Equation Md + Ndy =, where M = M(, y), N = N(, y). If the diff. Eq. Md + Ndy = is homogeneous then = is an integrating factor, where M + Ny and M = M(, y), N = N(, y) 5. If the diff. Eq. Md + Ndy = is of type f (, y)yd + f (, y)dy = then = is an integrating factor, where M - Ny. M N 6. IF is a function of alone then is an integrating factor of N equation Md + Ndy = where M = M(, y), N = N(, y) N M 7. IF is a function of y alone then is an integrating factor of M equation Md + Ndy = where M = M(, y), N = N(, y) 8. Solve the linear diff. Equation + Py = Q, where P & Q are functions of only. 9. Solve the linear diff. Equation + P = Q, where P & Q are functions of y only.. Eplain the method of solving the diff. Equation F(, y, p ) =, which is solvable for p, where p =.. Eplain the method of solving the diff. Equation F(, y, p ) =, which is solvable for y, where p =.. Eplain the method of solving the diff. Equation F(, y, p ) =, which is solvable for, where p =.
Q- Marks Solve the following differentials equations. sec tany d + sec y tan dy =. y sec d + (y+7) tan dy =. = +ycosy. (y- a(y + 5. ( -y )dy + (y +y )d = Solve the homogeneous diff. Eq. 6. ( +y )d y dy = 7. dy + (y -y)d = 8. ( +y-y ) dy + (y -y )d = 9. dy yd = d. = y(+y)/. ( -y ) d + y dy =.. = =. ( +y ) = y 5. ( + y cot/y ) dy y d = Solve the Non-homogeneous diff. Eq. 6. = 7. ( y + ) d + (y - )dy = 8. 9.. = = =
.... = = = = Solve the eact diff. Eq. 5. ( + y)d + ( + y - ) dy = 6. + = 7. ( + y - a ) d + ( y - b ) y dy = 8. ( + ) d + [ ( /y] dy = 9. (sec tan tany - e ) d + sec sec dy =. ( y y ) d + (y y + ) dy =. (e y + ) cos d + e y sin dy =. (sin cosy + e ) d + (cos siny + tany) dy =. [ - y] d + [y - ] dy =. [cos tany + cos( + y)] d + [sin sec y + cos( + y)] dy = Solve the Non-eact diff. Eq. 5. ( y y ) d ( y) dy = 6. ( 5y + 7y ) d + (5 7y) dy = 7. ( y + y + ) d ( y + 5y + ) ydy = 8. (y y ) d ( y - y ) dy = 9. ( + y) yd + ( - y) dy =. (y siny + cosy) yd + (y siny cosy) dy =. y(y + ) d + ( + y + y ) dy =. (y + y ) yd + (y - y ) dy =. (/+y) d + (/y-) dy =. ( y + y + y) yd + ( y - y + y) dy = 5. ( + y ) d y dy = 6. ( y + y + ) yd + ( y - y + ) dy =
7. ( + y) yd + ( - y) dy = 8. (y + y) d + ( y + + y ) dy = 9. (y + y) d + (y + y ) dy = 5. ( - y ) d + y dy = 5. ( y + y) d + ( y - ) dy = 5. ( y + y ) d + (/ + y ) dy = 5. ( e my ) d + m y dy = 5. ( + y + ) d + y dy = 55. ( + y + ) d + y dy = 56. ( - y ) d + y dy = 57. ( + y ) d + y dy = 58. (y + y y + 6) d + ( + y - ) dy = 59. y ( + y + ) d + (y - ) dy = 6. (7 y + y + ) d + ( + y) dy = Solve the Linear diff. Eq. 6. 6. y = e + y = 5 6. sin + y = cot 6. + y + y = 65. y + y = 66. ( y - y) dy = d 67. y - = y 68. = ( y) 69. = ( + y - 7) 7. cos + y sin = sin cos
Solve the following diff. Eq. for, y, p 7. p 5p + 6 = 7. p /p = /y y/ 7. p(p + y) = (+ y) 7. p(p - y) = (+ y) 75. p 7p + = 76. y = a/p + p 77. y = + p 78. y + logp = 79. y = p + p 8. y p = f(p ) 8. y = p + p y 8. p yp + y = 8. y = p + 6y p 8. y = p + y p 85. yp + ( + y + y )p + ( + y) = 86. y + log p = 87. y = ( + p) + p 88. y logy = yp + p 89. p = m + np
Que. Marks Write the definition of following. Homogeneous differential equation. Non- homogeneous differential equation. Eact differential equation. Linear differential equation 5. Bernaoll s differential equation 6. Claraut s differential equation Find the integrating factor of the following differential equation 7. ( + y ) d + ( - ) dy = 8. + y = 9. - tany = ( + ) e secy. ( cos) + ( sin + cos) y =. = y - y. (y + y) d + ( y + + y )dy =. ( + y + ) d + ydy =. (y + y) d + (y + y ) dy = Multiplying by appropriate integrating factor, make following diff. Eq. Eact. 5. ( y + ) yd + ( y ) dy = 6. ( y + y + ) yd + ( y y + ) dy = 7. (y y ) d - ( y - y ) dy = 8. ( + y ) d y dy = 9. (7 y + y ) d + ( + y) dy =. ( + y + ) d + y dy =
Write the appropriate answer of the following, where P & Q are functions of only.. The diff. Eq. + Py dy = Q is --- A) Linear D. E. B) Bernaoll s D. E. C) Eact D. E. D) Not eact D. E.. The diff. Eq. ( + y ) = y is --- A) Linear D. E. B) Homogeneous D. E. C) Bernaoll s D. E. D) Non- homogeneous D. E.. The diff. Eq. ( + y) yd + ( - y) dy = A) Not eact D. E. B) Clairaut s D. E. C) Linear D. E. D) Non- homogeneous D. E.. The diff. Eq. + y = is --- A) Not eact D. E. B) Clairaut s D. E. C) Linear D. E. D) Homogeneous D. E. 5. The diff. Eq. y = p + is --- A) Non-homogeneous D. E. B) Clairaut s D. E. C) Bernaoll s D. E. D) Homogeneous D. E.
Unit V Differential Equations Q-. Questions - Marks. Let f( D)y = X be the L.D.E. If = with constant coefficient. Then i) f ( D)y = is called --- ii) f (D ) = is called ---. If m, m, mn are n distinct real roots of A.E. f(d) = then G.S. of the equation f(d)y = is ---. If m = m two root of f(d) =, then C. F. of f(d)y = is ---. If m = α + iβ and m = α iβ are the comple roots of the f(d) =, then G.S. of f(d)y = is --- 5. If f ( D) = ( D m)( D m) ( D m n ), then the G.S. of the L.D.E. f(d)y = is ---. 6. If 7. If d y dy d d + + y = e,then what is its complementary function? f ( D ) is polynomial in D with constant coefficients and F(-a) then i) cos? ii) sin? 8. If d D = and f(d) is a polynomial in D with constant coefficients then d a i) e V =? f( D) ii) V =? f( D) where V is function of.
9. i) cos( a) =? r ( D + a ) ii) sin( a) =? ( D + a ). Let ( ) cos D + y=, find P.I.. If d D = and f(d) is a polynomial in D with constant coefficients then. d?,. a i) e =? r ( D a) r a ii) If f ( D) = ( D a) φ( D) and φ( a), then e =? f( D) Q-. Define the following. Linear differential equation with constant co-efficients of order n.. Associated D.E. and Auillary equation.. Inverse Operator. Homogeneous Linear Differential equation of the order n.
Q-. Multiple choices. If d y dy d d + y = e is a linear differential equation,then C.F. is ---- a)( c + c ) e b c c e )( + ) c)( c + c ) e d) none of these. If ( ) D + D + D+ y= e is a linear differential equation then C.F. is a c c c e )( + + ) b c c c e )( + + ) c)( c + c + c ) e d) none of these. If ( D + D+ ) y= is a linear differential equation then C.F. is --- ae ) ( ccos + csin ) be ) + ( ccos + c sin ) ce ) ( ccos + icsin ) d) none of these. If ( ) cos D + y= is a linear differential equation then C.F. is ------ ac ) cos+ csin bc ) cos+ ic sin cc ) sin+ ic cos d) none of these 5. If ( D ) cos + = is a linear differential equation then P.I. is ------ sin a) sin b) sin c) cos d)
d y dy 6. If + y = is a homogeneous L.D.E.,then solution of L.D.E. d d is ------ ay ) = ce + ce z z by ) = ce + ce z z cy ) = ce + ce z z d) none of these 7. If ( D + ) y= cos is a linear differential equation then C.F. is ------- a)( c + c )cos + ( c + c )sin b)( c+ c) cos + ( c + c)sin c)( c + c ) cos + ( c + c )sin dnoneofthese ) 8. If d y + y = is a linear differential equation then G.S. is ------- d aa ) cos+ Bsin ba ) cos+ Bsin ca ) sin+ Bcos d) none of these 9. If ( D 6D ) y + = is a linear differential equation then G.S. is ------- ) ( cos + sin ) ae A B ) ( cos + sin ) be A B ) (cos + sin ) ce B d) none of these. If - + y = is a homogeneous L.D.E., then G.S. is ------ i) (c + c log ) ii) e iii) e z iv) z ez
Q-. Numerical Eamples Marks ) Solve d y dy 5 + 6y = d d d y dy ) Solve + y = d d ) Solve ) Solve 5) Solve 6) Solve 7) Solve 8) Solve 9) Solve ) Solve ) Solve ) Solve ) Solve ) Solve 5) Solve 6) Solve d d dy d d d + + = d y dy + 5 y = d d d y y d + = d y dy + + y = e d d d y dy + y = d d d y y d + = ( D 6D + 9 D) y= ( D + 8D + 6) y= ( D ) ( D + ) y= ( + ) = cos D y d y d d y dy + y = e d dy 6y = e cosh d d d y dy + y = e d d 5 d y dy + + y = e d d
d y 7) Solve 9y e d = + 8) Solve 9) Solve ) Solve ) Solve ) Solve ) Solve ) Solve 5) Solve 6) Solve 7) Solve 8) Solve 9) Solve ) Solve ) Solve d y dy 5 + 6y = d d d y d y dy + y = cosh d d d d y y ( e ) d = + d y dy + + y e = + d d d y 8y d + = + + d y dy + 5y = d d d y d y + dy y = + d d d d y d y dy d d d + 6 + 8y = e + d y d y + 8 + 6= cos d d d y a y cos d = d y a y sin sin d + = d y y cos d + = d y dy + + y = sine d d d y dy + y = sin d d ) Solve ( D 5D + 6 )y = e ) Solve ( + + 6) = + sinh D D y e
) Solve ( D + D + D+ ) y= e 5) Solve 6) Solve 7) Solve 8) Solve 9) Solve ) Solve ) Solve ) Solve ) Solve ) Solve 5) Solve 6) Solve 7) Solve 8) Solve 9) Solve 5) Solve 5) Solve 5) Solve 5) Solve 5) Solve 55) Solve 56) Solve 57) Solve ( D 5D + 8D ) y= e ( ) D D+ y= e ( + ) = sinh D D y ( ) = cosh D Dy ( D 5D + 8D ) y= e + e ( D + D + D) y= ( D + D+ ) y = ( D D ) y= 9e ( D + D + D) y= + + 8 ( D D+ ) y= 8( + e ) ( + ) = 9 + 6 D D y ( + ) = cos + sin D D y ( + ) = sin D D D y ( + ) = sin D Dy ( + ) = cos D y ( ) = cos cos D y ( D + ) y= sin+ e + ( + ) = cos D Dy ( ) sin D y= ( ) cos D + y= ( D D 6 D) y= cos+ ( + ) = cosh + sin D D D y ( D D+ ) y= e
58) Solve ( D D+ ) y= e cos 59) Solve 6) Solve 6) Solve 6) Solve 6) Solve 6) Solve 65) Solve 66) Solve 67) Solve 68) Solve 69) Solve 7) Solve 7) Solve 7) Solve 7) Solve 7) Solve 75) Solve 76) Solve 77) Solve 78) Solve 79) Solve 8) Solve ( 6 + ) = sin D D y e ( ) cos D D+ y= e D D+ y = e ( ) ( ) = cos D y ( ) = sin D y ( ) = cos cosh D y ( ) = cos D y e ( D + ) y = e sin+ e sin ( D 7D 6) y= e ( + ) ( D D D + D+ ) y= e ( ) = sinh D y ( D ) y= e ( ) cos D + y= ( + ) = sin D y ( ) = cos D y ( + ) = cos D y ( + + ) = cos D D y ( + + ) = sin D D y ( D + Dy ) = ( + e) ( D + 5D+ 6) = e sec (+ tan ) ( + ) = sin D D e ( 9 8) e D D+ y = e
8) Solve ( D + D + )y = sin e 8) Solve ( D + D + )y = 8) Solve ( D + )y = tan 8) Solve ( D + D + )y = sin e - 85) Solve ( D - D + )y = e cos 86) Solve ( D - )y = ( + e - ) - 87) Solve ( D + D - )y = 88) Solve ( D - D + )y = 89) Solve ( D / D + / )y = (/ ) Log 9) Solve ( D - D - )y = Log 9) Solve ( D - D + 5 )y = sin (Log) 9) Solve [( + ) D ( + ) D - ]y = 6 9) Solve [( + ) D + ( + ) D + ]y = cos [ Log ( + ) ] 9) Solve ( D + D + )y = e 95) Solve [( - ) D + ( - ) D - ]y = 96) Solve [( + ) D + ( + ) D - 6 ]y = + + 97) Solve [( + ) D + (+ ) D + ]y = sin[ log(+) ] 98) Solve [( + ) D ( + ) D + 6 ]y = log ( + ) 99) Solve [( + ) D ( + ) D + ]y = +
Q-5. Theory Questions 6 Marks ) If d D = and f(d) is a polynomial in D with constant coefficients,then d Prove that ) Prove that, r a a e e = r ( D a) r! r a a e r Hence e =, if f ( D) = ( D a) φ( D) & φ( a) f( D) r! φ( a) ) If f ( D ) is polynomial in D with constant coefficients and f ( a ) cos( a + b) then prove that cos( a + b) = f( D ) f( a ) ) If f ( D ) is polynomial in D with constant coefficients and f ( a ) then sin( a + b) sin( a + b) = f ( D ) f( a ) 5) r r ( ) r Prove that cos a = cos a, r N r r + ( D + a ) r!( a) 6) r r ( ) r Prove that sin a = sin a, r N r r + ( D + a ) r!( a) 7) If d D = and f(d) is a polynomial in D with constant coefficients,then d Prove that a a e V = e V, f( D) f( D+ a) where V is a function of. 8) If d D = and f(d) is a polynomial in D with constant coefficients,then d Prove that V = f '( D) V, f( D) f( D) f( D) where Vis a function of. 9) Define a homogeneous linear differential equations & eplains the methods solving it.