{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

sol1 - ACM 95b/100b Solutions for Problem Set 1 Sean Mauch...

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

ACM 95b/100b Solutions for Problem Set 1 Sean Mauch [email protected] January 18, 2002 1 Problem 1 Problem. Find the general solution and plot some integral curves for each of the following ODEs. What is the behavior of the solution as t → ∞ ? 1. ty 0 - y = t 2 2. y 0 + y cos t = sin t cos t Solution. 1. We consider the differential equation ty 0 - y = t 2 . We can solve this differential equation by inspection. We guess a particular solution of the form y p = at 2 . By substituting into the differential equation we see that y p = t 2 . We see that y h = t is a homogeneous solution. Now we have the general solution of the differential equation. y = t 2 + ct We could also solve this problem by finding an integrating factor. y 0 - y t = t I = exp - Z 1 t d t = e - ln | t | = 1 | t | We choose I = 1 /t as the integrating factor. d d t y t = 1 y t = t + c y = t 2 + ct The solution, plotted in Figure 1, is unbounded as t → ∞ . 2. We consider the differential equation y 0 + y cos t = sin t cos t. 1

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

View Full Document
-2 -1 1 2 -2 -1.5 -1 -0.5 0.5 1 1.5 2 Figure 1: The integral curves of ty 0 - y = t 2 . Again we can solve this differential equation by inspection. We see that y p = sin t - 1 is a particular solution and y h = e - sin t is a homogeneous solution. The general solution of the differential equation is y = sin t - 1 + c e - sin t . We could also solve this problem by finding an integrating factor. I = exp Z cos t d t = e sin t d d t ( e sin t y ) = e sin t sin t cos t e sin t y = Z e sin t sin t cos t d t + c e sin t y = Z sin t e x x d x + c e sin t y = [( x - 1) e x ] sin t + c y = sin t - 1 + c e - sin t The solution, plotted in Figure 2, is bounded but does not have a limit as t → ∞ . 2 Problem 2 Problem. Solve the following linear initial value problems and in each case describe the interval on which the solution is defined. 1. y 0 + 2 xy = e - x 2 , y (0) = - 1 2. ty 0 + 2 y = t 2 - t + 1, y (1) = 1 / 2 Solution.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 10

sol1 - ACM 95b/100b Solutions for Problem Set 1 Sean Mauch...

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

View Full Document
Ask a homework question - tutors are online