Simplifying PDE

I want to ask you a question. For example I have an equation: $$u_{tt}-7u_{xx}-u_{x}=0 $$ To solve it I must first simplify it, right? I mean I must remove $u_x$. I suppose, that I must use next formulas: $$ u_{x} = e^{\lambda x + \mu t}(\lambda V + V_{x})$$ $$u_{xx} = e^{\lambda x + \mu t}(\lambda^{2} V + 2\lambda V_{x} + V_{xx})$$ $$u_{tt} = e^{\lambda x + \mu t}(\mu^{2} V + 2\mu V_{t} + V_{tt}) $$ Am I right? Will the primary conditions or\and boundary conditions change?

Solution 1:

You don't need those anzats, the equation is already amenable to a separation of variables via $u(t,x)=X(x)\cdot T(t)$. This substitution will give

$$ X(x)\cdot T''(t)- 7X''(x)\cdot T(t) - X'(x)\cdot T(t) = 0 $$ $$ X(x)\cdot T''(t)- (7X''(x) + X'(x))T(t)=0 $$ $$ \frac{T''(t)}{T(t)}=\frac{7X''(x) + X'(x)}{X(x)}=k,\hspace{5mm}k\in\mathbb{R} $$

Solve these single variable DE's with the help of your original BC/IC's.