Find the equation by applying the Laplace transform.

${y}^{\left(4\right)}-y=\mathrm{sin}ht$

$y(0)={y}^{\prime}(0)=y"(0)=0$

${y}^{\u2034}(0)=1$

melodykap
2021-02-10
Answered

Find the equation by applying the Laplace transform.

${y}^{\left(4\right)}-y=\mathrm{sin}ht$

$y(0)={y}^{\prime}(0)=y"(0)=0$

${y}^{\u2034}(0)=1$

You can still ask an expert for help

avortarF

Answered 2021-02-11
Author has **113** answers

Step 1

The given equation is,

${y}^{4}-y=\mathrm{sinh}t\dots \left(1\right)$

The given initial condition is,

$y\left(0\right)=1\dots \left(2\right)$

$y\prime \left(0\right)=1\dots \left(3\right)$

$y{}^{\u2033}\left(0\right)=1\dots \left(4\right)$

$y{}^{\u2034}\left(0\right)=1\dots \left(5\right)$

The Laplace transform is given as,

$L\left\{{F}^{n}\left(t\right)\right\}={s}^{n}L\left\{F\left(t\right)\right\}-{s}^{n-1}F\left(0\right)-{s}^{n-2}F\prime \left(0\right)-\dots -{F}^{n-1}\left(0\right)$

$L\{\mathrm{sin}hat\}=\frac{a}{{s}^{2}-{a}^{2}}$

Step 2

The given equation (1) is,

${y}^{4}-y=\mathrm{sinh}t$

Taking Laplace transform on both sides,

$L\{y{}^{\u2057}-y\}=L\{\mathrm{sin}ht\}$

$s}^{4}L\left\{y\right\}-{s}^{3}y\left(0\right)-{s}^{2}y\prime \left(0\right)-sy{}^{\u2033}\left(0\right)-y{}^{\u2034}\left(0\right)-L\left\{y\right\}=\frac{1}{{s}^{2}-1$

On putting the values of equation (2), (3), (4) & (5) in the above equation,

$s}^{4}L\left\{y\right\}-{s}^{3}\left(1\right)-{s}^{2}\left(1\right)-s\left(1\right)-\left(1\right)-L\left\{y\right\}=\frac{1}{{s}^{2}-1$

$s}^{4}L\left\{y\right\}-{s}^{3}-{s}^{2}-s-1-L\left\{y\right\}=\frac{1}{{s}^{2}-1$

${s}^{4}L\left\{y\right\}-L\left\{y\right\}=\frac{1}{{s}^{2}-1}+{s}^{3}+{s}^{2}+s+1$

$({s}^{4}-1)L\left\{y\right\}=\frac{1+{s}^{5}-{s}^{3}+{s}^{4}-{s}^{2}+{s}^{3}-s+{s}^{2}-1}{{s}^{2}-1}$

$({s}^{4}-1)L\left\{y\right\}=\frac{{s}^{5}+{s}^{4}-s}{{s}^{2}-1}$

The given equation is,

The given initial condition is,

The Laplace transform is given as,

Step 2

The given equation (1) is,

Taking Laplace transform on both sides,

On putting the values of equation (2), (3), (4) & (5) in the above equation,

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