smileycellist2

2021-02-08

Use the Laplace transform to solve the following initial value problem:
$2y"+4{y}^{\prime }+17y=3\mathrm{cos}\left(2t\right)$
$y\left(0\right)={y}^{\prime }\left(0\right)=0$
a)take Laplace transform of both sides of the given differntial equation to create corresponding algebraic equation and then solve for $L\left\{y\left(t\right)\right\}$b) Express the solution $y\left(t\right)$ in terms of a convolution integral

Step 1
Given differntial equation is,
$2y"+4{y}^{\prime }+17y=3\mathrm{cos}\left(2t\right)$
$y\left(0\right)={y}^{\prime }\left(0\right)=0$
$L\left[y"\right]={s}^{2}L\left\{y\left(t\right)\right\}-sy\left(0\right)-{y}^{\prime }\left(0\right)$
$L\left[{y}^{\prime }\right]=sL\left\{y\left(t\right)\right\}-y\left(0\right)$
$L\left[\mathrm{cos}\left(at\right)\right]=\frac{s}{{s}^{2}+{a}^{2}}$
Taking Laplace transform of equation (1),
$2L\left[y"\right]+4L\left[{y}^{\prime }\right]+17L\left[y\right]=3L\left[\mathrm{cos}\left(2t\right)\right]$
${s}^{2}L\left\{y\left(t\right)\right\}-sy\left(0\right)-{y}^{\prime }\left(0\right)+4\left[sL\left\{y\left(t\right)\right\}-y\left(0\right)\right]+17L\left\{y\left(t\right)\right\}=3L\left\{\mathrm{cos}\left(2t\right)\right\}$
${s}^{2}Ly\left(t\right)+4sLy\left(t\right)+17Ly\left(t\right)=3x\frac{s}{{s}^{2}+4}$
$\left({s}^{2}+4s+17\right)L\left\{y\left(t\right)\right\}=\frac{3s}{{s}^{2}+4}$
$a\right)\therefore L\left\{y\left(t\right)\right\}=\frac{3s}{{s}^{2}+4}×\frac{1}{{s}^{2}+4s+17}$
$b\right)y\left(t\right)={\int }_{0}^{t}\left[3\mathrm{cos}\left(3w\right)\right]×\left[{e}^{-2t}\cdot \mathrm{sin}\sqrt{13}w\right]dw$
This is required Laplace transform.

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