Exercise: Solving an IVP Problem with Laplace Transforms: Difference between revisions

From Class Wiki
Jump to navigation Jump to search
Main content
 
(No difference)

Latest revision as of 00:36, 29 January 2010

Author

John Hawkins

Problem Statement

Solve the following initial value problem using Laplace Transforms:

y4y=12x,y(0)=4,y(0)=1.


Note: This problem was solved by Zill without the use of Laplace Transforms.<ref>Dennis G. Zill, A first course in Differential Equations, 8th ed., Int. ed (Belmont, CA: Thomson Learning, 2005), 128.</ref>

Solution

Given the initial ODE

y4y=12x


we take the Laplace transform of both sides

{y4y}={12x}


Using the transforms displayed in Laplace Transform, we find this to be

[s2Y(s)sy(0)y(0)]4Y(s)=12s2


which, with initial values substituted, gives

(s24)Y(s)4s1=12s2


Hence,

Y(s)=12s2+4s+1s24


=12+4s3+s2s44s2


=12+s2+4s3s2(s2)(s+2)


Using a calculator to expand this, we have


Y(s)=1s+2+3s23s2


And therefore, using the equations on Laplace Transform to perform an inverse Laplace transform, we have our solution:

y(x)=e2x+3e2x3x


This equation matches that found by Zill, providing confirmation of a correct solution.

References

<references />

Reviewed By

Read By

Comments