UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS

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1 VAN/SURNAME: UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS VOORNAME/FIRST NAMES: WTW 162 DYNAMICAL PROCESSES EKSAMEN / EXAMINATION STUDENTENOMMER/STUDENT NUMBER: HANDTEKENING/SIGNATURE: November 2013 TYD/TIME: 90 min PUNTE / MARKS: 40 Eksterne Eksaminator/External Examiner: Interne Eksaminatore/Internal Examiner: Prof A Harding Prof R Anguelov PUNTE / MARKS LEES DIE VOLGENDE INSTRUKSIES 1. Die vraestel bestaan uit bladsye 1 tot 9 (vrae 1 tot 7). Kontroleer of jou vraestel volledig is. 2. Doen alle krapwerk op die teenblad. Dit word nie nagesien nie. 3. As jy meer as die beskikbare ruimte vir n antwoord nodig het, gebruik dan ook die teenblad en dui dit asseblief duidelik aan. 4. Geen potloodwerk of enige iets wat in rooi ink gedoen is, word nagesien nie.. As jy korrigeerink ( Tipp-Ex ) gebruik, verbeur jy die reg om te kla oor werk wat nie nagesien is nie of wat verkeerd nagesien is. Outeursreg voorbehou READ THE FOLLOWING INSTRUCTIONS 1. The paper consists of pages 1 to 9 (questions 1 to 7). Check whether your paper is complete. 2. Do all scribbling on the facing page. It will not be marked. 3. If you need more than the available space for an answer, use the facing page and please indicate it clearly. 4. No pencil work or any work in red ink will be marked.. If you use correcting fluid ( Tipp-Ex ), you lose the right to question the marking or to indicate work that had not been marked. Copyright reserved

2 Vraag 1 Question 1 Bepaal die eksplisiete algemene oplossing van die volgende vergelyking 2xy + y = 10 x Find the explicit general solution of the following equation 2

3 Vraag 2 Question 2 Bepaal die explisiete partikuliere oplossing van die volgende beginwaarde-probleem Find the explicit particular solution of the following initial value problem dy dx = 4x3 y y, y(1) = 3 3

4 Vraag 3 Question 3 n Oorverhittende enjin wat teen 270 o C is word afgeskakel en gelaat om af te koel. Na een uur is die temperatuur 90 o C. As die temperatuur van die omgewing 2 o C is, hoe lank sal dit neem vir die enjin om af te koel tot 4 o C? An overheating car engine which is at 270 o C is turned off and allowed to cool. After one hour the temperature is 90 o C. If the temperature of the surrounding is 2 o C, how long does it take for the engine to cool to 4 o C? 4

5 Vraag 4 Question 4 A n 1000 liter tenk bevat aanvanklik 20 liters soutwater wat 2 kilogram sout bevat. Soutwater wat 10 gram sout per liter bevat vloei die tenk binne teen n tempo van 10 liter per minuut, en die goedgemengde soutwater in die tenk vloei uit teen n tempo van liter per minuut. Hoeveel sout sal daar in die soutwater wees as die tenk vol is? A 1000 liter tank initially contains 20 liters of brine containing 2 kilograms of salt. Brine containing 10 grams of salt per liter enters the tank at a rate of 10 liters per minute, and a well mixed brine in the tank flows out at a rate of liters per minute. How much salt will there be in the brine when the tank is full? 6

6 Vraag Question Beskou die logistiese vergelyking met oesting dy dt = 0.y(1 0.2y) h Consider the logistic equation with harvesting.1 Vir watter waardes van h het die model (i) twee ewewigte, (ii) een ewewig, (iii) geen ewewig?.1 For what values of h does the model have (i) two equilibria, (ii) one equilibrium, (iii) no equilibria? 2.2 Skets n verteenwoordigende versameling oplossings vir die gevalle waar daar (i) twee ewewigte, (ii) een ewewig, (iii) geen ewewig is..2 Sketch a representative set of solutions for the cases when there is (i) two equilibria, (ii) one equilibrium, (iii) no equilibria. 3 6

7 Vraag 6 Question 6 Beskou die bevolkingsmodel: Consider the population model: dp dt = P P 2 ) + P 6.1 Bepaal die ewewig(te) van die model and stel vas met motivering die stabiliteitseienskappe (stabiel or unstabiel, aaantrekkend of afstotend of nie een van die twee nie). 6.1 Find the equilibrium(s) of the model and establish with motivation its (their) stability properties (stable or unstable, attracting or repelling or neither). 6.2 Skets op n tyddiagram n verteenwoordigende versameling oplossings wat die eienskappe van die ewewigte wat afgelei is in 6.1 aandui. 6.2 Sketch on a time diagram a representative set of solutions indicating the properties of the equilibrium(s) derived in

8 6.3 Gestel dat die bevolking onderhewig is aan konstante poging oesting. Bepaal die Maksimum Volhoubare Opbrengs (MVO) 6.3 Suppose that the population is subjected to constant effort harvesting. Find the Maximum Sustainable Yield (MSY) 4 8

9 Vraag 7 Question 7 Gegee die volgende stelsel van differensiaalvergelykings x = xy 2x y = x 2 18y Given is the following system of differential equations Bepaal al die ewewigte van die stelsel Find all equilibria of the system 3 9

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