Separation of variables - please help!

Question

I am doing a question on Separation of variables:

My doubt here is that once the partial derivatives have been found and have been substituted back into the original PDE I am left with a ''+ U'' and I am not sure where to go from here.

Thank you

Answer 1

If you set

$$ U(x,t) = X(x) \cdot T(t) $$

you'll get:

$$ X(x) \cdot T'(t) + X(x) \cdot T(t) = X''(x) \cdot T(t) $$

Now, divide by X(x)·T(t) to get:

$$ \frac{T'(t)}{T(t)} + 1 = \frac{X''(x)}{X(x)} $$

And each side must be independent of any parameter, so:

$$ \frac{T'(t)}{T(t)} + 1 = c $$

and

$$ \frac{X''(x)}{X(x)} = c $$

being c the same constant in both cases.

Answer 2

Isn't the substitution just $U(x,t)=f(x)g(t)$, so you turn $U_t +U =U_{xx}$ into $fg' + fg = f'g$, which can be separated as $g'/g + 1 = f'/f$ ?