Find the general solution to these partial differential equations?

Question

Find the general solution to these partial differential equations?

(b) $\dfrac{\partial u}{\partial y} - u = e^{x-y} $

(c) $u_{xx} + y^2u = 0 $

Any help would be appreciated, thanks!

Answer 1

$$ \partial_y u - u = \mathrm{e}^{x-y} $$ $$ u\mathrm{e^{-y}} = \int \mathrm{e}^{x-y}\mathrm{e}^{-y}dy + g(x) = \int \mathrm{e}^{x}\mathrm{e}^{-2y}dy + g(x) $$ $$ u(x,y) = -\mathrm{e}^{y}\frac{1}{2}\mathrm{e}^{x}\mathrm{e}^{-2y} + g(x)\mathrm{e}^{y} = -\frac{1}{2}\mathrm{e}^{x-y} + g(x)\mathrm{e}^{y} $$ where $g(x)$ is a function determined from conditions of the problem.

the next you can do as a similar fashion, by integrated as you would do for the 1-D problem but treat constants as that of the variable we are not integrating with, and it should lead to $$ u(x,y) = A(y)\cos\left(yx\right) + B(y)\sin\left(yx\right) $$