Exercise: Solving an IVP Problem with Laplace Transforms

Author

John Hawkins

Problem Statement

Solve the following initial value problem using Laplace Transforms:

$$\begin{gathered} y^{\prime \prime }-4y=12x, \\ y(0)=4, \\ {y}^{\prime }(0)=1. \end{gathered}$$

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

$y^{\prime \prime }-4y=12x$

we take the Laplace transform of both sides

${ℒ}\{y^{\prime \prime }-4y\}={ℒ}\left\{12x\right\}$

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

$[s^{2}Y(s)-sy(0)-y^{\prime }(0)]-4Y(s)=\frac{12}{s^2}$

which, with initial values substituted, gives

$\Rightarrow (s^2-4)Y(s)-4s-1=\frac{12}{s^2}$

Hence,

$Y(s)=\frac{\frac{12}{s^2}+4s+1}{s^2-4}$

$=\frac{12+4s^3+s^2}{s^4-4s^2}$

$=\frac{12+s^2+4s^3}{s^2(s-2)(s+2)}$

Using a calculator to expand this, we have

$Y(s)=\frac{1}{s+2}+\frac{3}{s-2}-\frac{3}{s^2}$

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

$y(x)=e^{-2x}+3e^{2x}-3x$

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

References

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