GSW Problem Set 1 Answers

GSW Problem Set 1 Answers - EA4 Workshop questions with...

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Unformatted text preview: EA4 Workshop questions with answers - set 1 - week of 9/29/08 1. Consider the equation dy dt = y n , y (0) = 1 , (1) where n is a positive integer. (a) Find the solution for all values of n 1 . Explain why you have to use a different procedure for n = 1 than for n > 1 . Answer- the equation is separable so we can write dy y n = dt. Consider first n > 1 . Integrating both sides we have 1 1- n y 1- n = t + c, where c is an integration constant. Now use the initial condition to obtain c =- 1 n- 1 . Next, multiply both sides of the equation by 1- n to get y 1- n = 1- ( n- 1) t. Inverting the fraction we get y = 1 (1- ( n- 1) t ) 1 n- 1 . (2) When n = 1 the integrals become dy y = dt Integrating both sides now gives natural logs rather than a polynomial term in y . This is why the situation for n = 1 is different from the n > 1 case. We get ln( | y | ) = t + c. Evaluating c from the initial condition, we get c = 0 . Raising both sides to the power e gives y = e t . (3) (b) Show that for n 2 the solution blows up at a finite time. Answer- the denominator in equation (2) becomes zero when t = 1 / ( n- 1) . This is the blowup time. For n = 1 , it is clear from (3) that the solution is defined for all t although it approaches infinity for large t . 1 (c) Show that the blowup time gets smaller as n increases. Explain why this is so. Answer- From the differential equation (1) for any fixed value of y , dy/dt gets bigger and bigger as n increases. Thus, you expect the solution to grow more rapidly as n increases. This means that the blowup occurs earlier. This is verified by the solution which shows that the blowup time is 1 /...
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This note was uploaded on 09/08/2010 for the course GEN_ENG 205 taught by Professor Taflove during the Fall '08 term at Northwestern.

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GSW Problem Set 1 Answers - EA4 Workshop questions with...

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