Here's the modified heat equation i was given : $\frac{\mathrm{d}u}{\mathrm{d}t} = \frac{\mathrm{d}^2u}{\mathrm{d}x^2} + u$ , $L = \pi$, $BC = 0$ at both ends ( $x = 0$ and $\pi$ )
IC: $u (x,0) = x(\pi - x)$
Here's my attempt at constructing ODEs for $F(x)$ and $G(t)$
let $u(x,t) = F(x)G(t)$
$\implies G^{\prime}(t)F(x) = F^{\prime \prime}(x)G(t) + F(x)G(t)$
$\cfrac{G^{\prime}(t)}{G(t)} = \cfrac{F^{\prime \prime}(x)+F(x)}{F(x)} = a$, where $a$ is the separation constant
solving for $F(X)$:
$F^{\prime \prime}(x) - F(x)a + F(x) = 0$
$F^{\prime \prime}(x) + F(x)(1-a) = 0$
This is where i get stuck, is it safe to assume that since a is some constant, then $(1 - a)$ is also some arbitrary constant as well? , say $p^2$
If so, then the rest of steps in solving for F(x) seems to be simple enough
For, $p^2 \gt 0$
Which should eventually just give me $F(x) = \sin(nx)$, where $n = R/0$
If so, how does this affect me trying to find the general solution for $G(t)$?
Trying to solve for $G(t)$:
$\cfrac{G^{\prime}(t)}{G(t)} = a$
$G^{\prime}(t) - (a)G(t) = 0$
Note that $K=-\partial _{x}^{2}$ has real, non-negative, discrete spectrum, \begin{equation*} K=\sum_{n}\lambda _{n}|v_{n}><v_{n}| \end{equation*} and \begin{equation*} u(t)=\exp [Kt]u(0)=\sum_{n}e^{\lambda _{n}t}<v_{n}|u(0)>v_{n}(x) \end{equation*} so you see that your ansatz only works if $<v_{n}|u(0)>$ vanishes for all but one $n$.