Solving $xy'+y=x^{k}$

Question

Find a solution to: $$xy'+y=x^{k}$$ Where $k>0$, and on the assumption that the transforms of $f$ and $f'$ exist.

I understand that we can take the Laplace of all of the terms and then find the inverse Laplace transform to get a solution for $y$.

I am unsure of how to find the Laplace of the first term. Can someone show me how its done? It has to be using Laplace transforms !

Answer 1

Hint: you can use the fact that

$$\mathcal{L}\{f'(t)\} = s\mathcal{L}\{f(t)\}-f(0)$$

and

$$\mathcal{L}\{tf(t)\} = -\frac{d}{ds}\mathcal{L}\{f(t)\}$$

to get that

$$\mathcal{L}\{tf'(t)\} = -\frac{d}{ds}\mathcal{L}\{f'(t)\} = -\frac{d}{ds}\big(s\mathcal{L}\{f(t)\}-f(0)\big) = - \mathcal{L}\{f(t)\} - s\frac{d}{ds}\mathcal{L}\{f(t)\}$$

but you really don't need to resort to Laplace in this case.

Answer 2

Note that $$ (xy(x))'=xy'(x)+y(x) $$ so that your differential equation can be solved by one simple integration.