Reginald Metcalf

2022-01-21

Solve by using the Laplace Transform Method?

Laura Worden

Given first order differential equation

$y\left(0\right)=3$
claim- to check whether the differential equation can be solved using laplace transform or not
solution
applying laplace both sided
$L|{y}^{\prime }|+L|y|=L|6{e}^{{t}^{2}}|$
as we know
$L|{y}^{\prime }|=sy\left(s\right)-y\left(0\right)$
$L|y|=y\left(s\right)$
equation (i) becomes,
$sy\left(s\right)-y\left(0\right)+y\left(s\right)=6L\left[{e}^{{t}^{2}}\right]$
$L\left[{e}^{{t}^{2}}\right]=\frac{1}{2}\sqrt{\pi }{e}^{\frac{-{s}^{2}}{4}}erf\left(\frac{\pi }{2}\right)-\frac{1}{2}i\sqrt{\pi }{e}^{\frac{-{s}^{2}}{4}}$
thus, we get that the first order differential equation cannot be solved using laplace transform, because the laplace transform of right hand side is not in the form of standard function.

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