I am asked to find the general solution $f(x, y)$ of the partial differential equation:
$\frac{\partial ^2 f}{\partial x \partial y}=e ^ {x+2y}$
I know these are relatively easy to solve, I haven't done them in a while and have forgotten how to go about solving them, I haven't yet found an good internet source that explains them straightforwardly.
To attempt a solution, I first found the integral,
$\int e^x e^y dx=e^x e^y +g(y)$
Next, integrating this with respect to $y$,
$\int (e^x e^y +g(y)) \space dy$
solving this becomes,
$ = e^x e^y +yg(y) +h(x)$
Is my reasoning correct? If I integrate a partial derivative with respect to $x$, will the constant become $g(y)$ and if I integrate a partial derivative with respect to $y$, will the content become $h(x)$?
Best Answer
The methodology in the posted question was correct and gives a way forward. Its implementation had some mistakes which we resolve here.
We begin with
$$\frac{\partial^2 f(x,y)}{\partial x\partial y}=e^{x+2y} \tag 1$$
and integrate $(1)$ with respect to $x$ to obtain
$$\frac{\partial f(x,y)}{\partial y}=e^{x+2y}+C_1(y) \tag 2$$
where $C_1(y)$ is an integration constant.
Next, we integrate $(2)$ with respect to $y$ and obtain
$$f(x,y)=\frac12 e^{x+2y}+\int C_1(y)\,dy+C_2(x)$$
where $C_2(x)$ is a second integration constant.
Finally, labeling $g(y)=\int C_1(y)\,dy$ and $h(x)=C_2(x)$ yields the general result
$$\bbox[5px,border:2px solid #C0A000]{f(x,y)=\frac12 e^{x+2y}+g(y)+h(x)}$$