In an PDE, if these kind of equation: $\frac{\partial \:u\left(x,\:t\right)}{\partial \:t}+\frac{\partial ^2u\left(x,\:t\right)}{\partial \:x\partial \:t}+\frac{\partial ^2u\left(x,\:t\right)}{\partial \:x^2}=0 $
How can i deal with it?
Am i doing right??
You are going in the right direction. Starting with $X(x)T'(t)+X'(x)T'(t)=-X''(x)T(t)$ we can obtain $$\frac{T'(t)}{T(t)}=-\frac{X''(x)}{X(x)+X'(x)}=\lambda$$ where $\lambda$ is some constant. We can do this because two functions with different variables can be equal only when they are constants. Thus, we obtain two equations to find $T(t)$ and $X(x)$:
$\frac{T'(t)}{T(t)}=\lambda$ and $-\frac{X''(x)}{X(x)+X'(x)}=\lambda$