# Find the solution of the following differential equation by Laplace transforms: y'''- 5y" + 7y’-3y =20sin(t) , y(0)=y'(0)=0 , y"(0)=-2

Find the solution of the following differential equation by Laplace transforms:
${y}^{‴}-5y"+7{y}^{\prime }-3y=20\mathrm{sin}\left(t\right),y\left(0\right)={y}^{\prime }\left(0\right)=0,y"\left(0\right)=-2$
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Step 1
Consider the following initial value problem and apply the Laplace transform on both sides:
${y}^{‴}-5y"+7{y}^{\prime }-3y=20\mathrm{sin}t$
$y\left(0\right)={y}^{\prime }\left(0\right)=0,y"\left(0\right)=-2$
${s}^{3}L\left\{y\left(t\right)\right\}-{s}^{2}y\left(0\right)-s{y}^{\prime }\left(0\right)-y"\left(0\right)-5\left\{{s}^{2}L\left\{y\left(t\right)\right\}-sy\left(0\right)-{y}^{\prime }\left(0\right)\right\}+7\left\{sL\left\{y\left(t\right)\right\}-y\left(0\right)\right\}-3L\left\{y\left(t\right)\right\}=\frac{20}{\left({s}^{2}+1\right)}$
${s}^{3}Y\left\{y\left(t\right)\right\}+2-5\left\{{s}^{2}L\left\{y\left(t\right)\right\}\right\}+7\left\{sL\left\{y\left(t\right)\right\}\right\}-3L\left\{y\left(t\right)\right\}=\frac{20}{\left({s}^{2}+1\right)}$
$\left({s}^{3}-5{s}^{2}+7s-3\right)L\left\{y\left(t\right)\right\}+2=\frac{20}{\left({s}^{2}+1\right)}$
$\left({s}^{3}-5{s}^{2}+7s-3\right)L\left\{y\left(t\right)\right\}=\frac{20}{\left({s}^{2}+1\right)}-2$
$L\left\{y\left(t\right)\right\}=\frac{20}{\left({s}^{2}+1\right)\left({s}^{3}-5{s}^{2}+7s-3\right)}-\frac{2}{\left({s}^{3}-5{s}^{2}+7s-3\right)}$
$=\frac{1-3s}{\left({s}^{2}+1\right)}+\frac{3}{\left(s+1\right)}-\frac{4}{\left(s-1{\right)}^{2}}$
$=\frac{1}{\left({s}^{2}+1\right)}+\frac{3}{\left(s+1\right)}-\frac{4}{\left(s-1{\right)}^{2}}-\frac{3s}{\left({s}^{2}+1\right)}$
Step 2
Apply the inverse Laplace transform:
$L\left\{y\left(t\right)\right\}=\frac{1}{\left({s}^{2}+1\right)}+\frac{3}{\left(s+1\right)}-\frac{4}{\left(s-1{\right)}^{2}}-\frac{3s}{\left({s}^{2}+1\right)}$
$\left\{y\left(t\right)\right\}={L}^{-1}\left[\frac{1}{\left({s}^{2}+1\right)}+\frac{3}{\left(s+1\right)}-\frac{4}{\left(s-1{\right)}^{2}}-\frac{3s}{\left({s}^{2}+1\right)}\right]$
$y\left(t\right)=\mathrm{sin}t+3{e}^{t}-4t{e}^{t}-3\mathrm{cos}t$