Partial derivatives of second order

Question

Find all functions $f:\mathbb{R}^2\rightarrow \mathbb{R}$ of class ${\cal C}^2$, such that:

$\frac{\partial^2f}{\partial x\partial y} = 0$

$\frac{\partial^2f}{\partial x^2} = \frac{\partial^2f}{\partial y^2}$

(Separate questions)

For the first one I prove that $f(x,y) = h(x)+g(y)$ for some $h,g\in{\cal C}^2$, but I can't determine a condition for $f$ in the second part.

Do you want both conditions to hold or are these two separate questions? — Michael Albanese, Oct 20 '13 at 04:44
Just to explain the answer of @MichaelAlbanese : making in the second equation the change of variables $\xi=x+y$, $\eta=x-y$, you can reduce it to the first. — Start wearing purple, Oct 20 '13 at 04:59

score 3 · Accepted Answer · answered Oct 20 '13 at 04:56

3

The second equation is the one-dimensional wave equation which has solutions $f(x, y) = g(x+y) + h(x-y)$; this form is usually obtained via d'Alembert's solution which is explained in the link.

answered Oct 20 '13 at 04:56

Michael Albanese

99,526

Partial derivatives of second order

1 Answers1