u_x + u_y = u

Use separation of variables to find product solutions for the given partial differential equation.

Question

u_x + u_y = u

Use separation of variables to find product solutions for the given partial differential equation.

anonymous · Answer

$$u _{x} + u_{y} = u$$

anonymous · Answer

I get to:
$$Y' - (1+\lambda)Y = 0$$
$$X' + \lambda X = 0$$

anonymous · Answer

using u = XY,

but I don't understand how things comes to

$$X = c_{1}e^{-\lambda x} , Y = c_{2} e^{(1 + \lambda) y}$$

anonymous · Answer

I think I'm forgetting something from ODE's...

kinggeorge · Answer

Well, if you have $X'=-\lambda X$, you can see that $$X=-\lambda \int Xdx$$Then is should be clear that you get $c_1e^{-\lambda x}$, since $e$ has that exact property. If things like this aren't immediately obvious, try putting $\lambda=1$ to see if that gets you anywhere.

And then you have the same thing for the $y$ case, just with $\lambda+1$.

anonymous · Answer

hmm

anonymous · Answer

ok I think I'm getting it. Would you mind if we do another similar problem?

kinggeorge · Answer

I can try. To be honest, I haven't done much with PDE's, although if it's similar it's probably fine.

anonymous · Answer

Ok, thanks so much by the way