I am reading a text about Laplace transform in solving differential equations. Seems that this method is so powerful. To practice how it works, I makeup a very simple problem(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{dy}{dt} = e^{wt}y[/tex]

This equation could be solved by variable seperation and then intergrate LHS and RHS. But I would like to check that Laplace transform works. Let's

[tex]Y = \mathcal{L}[y][/tex]

is the Laplace transform of y. Note that

[tex]\mathcal{L}[dy/dt] = sY(s) - y(0)[/tex]

and

[tex]\mathcal{L}[e^{wt}y] =Y(s-w)[/tex]

where s is the parameter for Laplace transform. Hence, in Laplace domain, above equation becomes

[tex]sY(s) = Y(s-w)[/tex]

(assuming y(0)=0) My question is : there is a shift in the variable s, so how to solve this equation to get Y?

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# Laplace transform

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