Exercise: Solving an IVP Problem with Laplace Transforms

Author

John Hawkins

Problem Statement

Solve the following initial value problem using Laplace Transforms:

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

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

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

we take the Laplace transform of both sides

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

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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