Question

Is this equation separable?
\[u_{xx}+u_{xt}+u_t=0\]
I simplify down to
\[X''(x)T(t)+X'(x)T'(t)+X(x)T'(t)=0\]

Which does not look separable to me.

Answer 1

Even if you divided by X(x)T(t)

\[\frac{X''(x)}{X(x)}+\frac{X'(x)T'(t)}{X(x)T(t)}+\frac{T'(t)}{T(t)}=0\]

Which is almost separable but I don't think you can do anything with the middle term.

Answer 2

Turns out it is:

\[X''(x)T(t)+X'(x)T'(t)+X(x)T'(t)=0\]
\[X''(x)T(t)+T'(t)[X'(x)+X(x)]=0\]
Divide by T(t)(X'(x)+X(x))
\[\frac{X''(x)}{X'(x)+X(x)}+\frac{T'(t)}{T(t)}=0\]

\[\frac{X''(x)}{X'(x)+X(x)}=-\frac{T'(t)}{T(t)}\]
\[\rightarrow \frac{X''(x)}{X'(x)+X(x)}=-\lambda , \frac{T'(t)}{T(t)}=-\lambda \]