{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 /...
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online