Differential Equations Practice Exams

Answers


Problem 1: Use variation of parameters to find the general solution to

displaymath182

First we need to solve the homogeneous equation y''' - y' = 0. Its characteristic equation is tex2html_wrap_inline192 . It is easy to see that its root are r=0,1,-1. Therefore we have tex2html_wrap_inline196 .
Second we need to find a particular solution using the variation of parameters technique. We have tex2html_wrap_inline198 , where u', v', and w' are solution to the system

displaymath206

Easy calculations give

displaymath208

Integration by parts, will give (you are encouraged to do it)

displaymath210

Hence we have

displaymath212

which implies that tex2html_wrap_inline214 once u, v, w are plugged into the formula giving tex2html_wrap_inline222 . Therefore the general solution is given by:

displaymath224

Problem 2. Find the solution to the initial value problem

displaymath183

where

displaymath184

Since g(t) is a step function, we need to use Laplace Transform to solve this problem. Once we attack the equation by tex2html_wrap_inline228 , we get

displaymath230

which implies tex2html_wrap_inline232 . Using the initial condition, we get

displaymath234

After easy calculations, we obtain

displaymath236

Since tex2html_wrap_inline238 , we deduce

displaymath240

Hence

displaymath242

In order to find y, we need to use the inverse Laplace transform. First we have

displaymath246

which implies

displaymath248

The hard part concerns the second term tex2html_wrap_inline250 . First let us find the inverse Laplace without the exponential. First we know that

displaymath252

Using the derivative formula, we get

displaymath254

Therefore, we have

displaymath256

Using the formula tex2html_wrap_inline258 , we get

displaymath260

Therefore, we have

displaymath262

Problem 3. Find the Laplace transform of

displaymath185

We have

displaymath264

Hence tex2html_wrap_inline266 . Using the formula tex2html_wrap_inline268 , we get

displaymath270

But tex2html_wrap_inline272 which implies

displaymath274


[Next Exam] [Calculus] [CyberExam]

S.O.S MATH: Home Page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour