Hi Suppose We have a partial equation like below :
$\displaystyle u_{xx} + u_{x} - 2u =0$
I am wondering how to solve that ? I have misunderstanding for partial functions...
Since the only derivatives are with respect to x, for this particular example, you can (almost) treat this as an ordinary differential equation. $\displaystyle u_{xx}- u_x+ 2u= 0$ is a linear equation with constant coefficients. Its characteristic equation is $\displaystyle r^2- r+ 2= 0$. The only change for the partial differential equation is that the "constants" in the general solution may be functions of whatever the other variables are.
The discriminant of the charactestic equation is <0, so that the roots are complex. If $\displaystyle \alpha \pm i\ \beta$ are the roots then the solution is...
$\displaystyle u= e^{\alpha\ x} (c_{1}\ \cos \beta x + c_{2}\ \sin \beta x)$ (1)
... where $\displaystyle c_{1}$ and $\displaystyle c_{2}$, as HalsofIvy has precised, are independent from x...
Marry Christmas from Serbia
$\displaystyle \chi$ $\displaystyle \sigma$