# Solve the given symbolic initial value problem and sketch a graph of the solution. y′′+y=3delta(t-(frac{pi}{2}) y(0)=0​ y'(0)=3

Solve the given symbolic initial value problem and sketch a graph of the solution.
$y\prime \prime +y=3\delta \left(t-\left(\frac{\pi }{2}\right)y\left(0\right)=0{y}^{\prime }\left(0\right)=3$
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Step 1
Consider the provided question,
Solving the given D.E by using the Laplace Transform.
Given that , $y\prime \prime +y=3\delta \left(t-\left(\frac{\pi }{2}\right)y\left(0\right)=0{y}^{\prime }\left(0\right)=3$
take the Laplace transform on both side.
$L\left(y"\right)+L\left(y\right)=3L\delta \left(t-\left(\pi /2\right)\right)$

${s}^{2}Y\left(s\right)-sy\left(0\right)-{y}^{\prime }\left(0\right)+Y\left(s\right)=3{e}^{-\frac{\pi }{2s}}$
${s}^{2}Y\left(s\right)-0-3+Y\left(s\right)=3{e}^{-\frac{\pi }{2s}}$
$Y\left(s\right)\left({s}^{2}+1\right)=3{e}^{-\frac{\pi }{2s}}+3$
$Y\left(s\right)=\frac{3{e}^{-\frac{\pi }{2s}}}{{s}^{2}+1}+\frac{3}{{s}^{2}+1}$
$L\left(y\right)=\frac{3{e}^{-\frac{\pi }{2s}}}{\left({s}^{2}+1\right)}+\frac{3}{\left({s}^{2}+1\right)}$
Step 2
Now, use the inverse Laplace transformation,
$L\left(y\right)=\frac{3{e}^{-\frac{\pi }{2}s}}{\left({s}^{2}+1\right)}+\frac{3}{\left({s}^{2}+1\right)}$
$y={L}^{-1}\left(\frac{3{e}^{-\frac{\pi }{2}s}}{\left({s}^{2}+1\right)}\right)+{L}^{-1}\left(\frac{3}{\left({s}^{2}+1\right)}\right)$

Where , H(t) is Heaviside step function
$y=H\left(t-\frac{\pi }{2}\right)\cdot 3\mathrm{sin}\left(t-\frac{\pi }{2}\right)+3\mathrm{sin}\left(t\right)$
Thus , solution of the given D.E. is ,
$y=3H\left(t-\frac{\pi }{2}\right)\cdot \mathrm{sin}\left(t-\frac{\pi }{2}\right)+3\mathrm{sin}\left(t\right)$
Step 3
The graph of the above solution is drawn as,
$y=3H\left(t-\frac{\pi }{2}\right)\cdot \mathrm{sin}\left(t-\frac{\pi }{2}\right)+3\mathrm{sin}\left(t\right)$