Laplace of the PDE $ x \frac{\partial(w)}{\partial(x)} + \frac{\partial(w)}{\partial(t)}=xt$.

Question

Question is find the Laplace transform of this equation: $$ x \frac{\partial(w)}{\partial(x)} + \frac{\partial(w)}{\partial(t)}=xt$$

Boundary conditions : $$ w(x,0) = 0 \qquad x \geq0 $$ $$w(0,t)=0 \qquad t \geq 0$$

I've got as far as $$ \frac{d(W)}{d(x)}+\frac{s}{x}W=\frac{1}{s^2}$$

I'm not even sure this is correct but even if it is, I dont know how to integrate this to get $W(x,s)$

Answer 1

$$\frac{dW}{dx}+\frac{s}{x}W=\frac{1}{s^2}$$

Following the hints in the comments, the solution of the equation is

$$W(x,s)=\dfrac{x}{s^2(s+1)}+c_1x^{-s}$$

But $W(0,s)=0$ and because it holds for any $s$

$$0=\dfrac{c_1}{x^s}\implies c_1=0$$

$$W(x,s)=\dfrac{x}{s^2(s+1)}$$

and

$$w(x,t)=x(e^{-t}+t-1)$$