Reginald Metcalf

2022-01-21

Solve by using the Laplace Transform Method?

${y}^{\prime}+y=6{e}^{{t}^{2}},\text{}\text{}\text{}y\left(0\right)=3$

Laura Worden

Beginner2022-01-21Added 45 answers

Given first order differential equation

${y}^{\prime}+y=6{e}^{{t}^{2}},\text{}\dots \left(i\right)$

$y\left(0\right)=3$

claim- to check whether the differential equation can be solved using laplace transform or not

solution

applying laplace both sided

$L\left|{y}^{\prime}\right|+L\left|y\right|=L\left|6{e}^{{t}^{2}}\right|$

as we know

$L\left|{y}^{\prime}\right|=sy\left(s\right)-y\left(0\right)$

$L\left|y\right|=y\left(s\right)$

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.

claim- to check whether the differential equation can be solved using laplace transform or not

solution

applying laplace both sided

as we know

equation (i) becomes,

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.