Quote by Gregg
I have forgotten about the function ## F(x,t) = \phi(x) G(t)## We have solutions of the form ##F_n(x,t) = (A_n \sin (n \pi x) + B_n \cos (n \pi x) ) c_3 e^{\lambda t} ##
Ah so we have
##F(x,0) = x^2, ## ##x\in [0,1] ##
So this series
##F(x,0) = \sum_{n=0}^{\infty} \left( A_n \sin (n \pi x) + B_n \cos (n \pi x) \right)= x^2 ##
## x \in [0,1] ##
So that's where Fourier comes in

You mean$$
\phi_n(x) = \left( A_n \sin (2n \pi x) + B_n \cos (2n \pi x) \right)$$ and $$
G_n(t) = e^{4n^2\pi^2 t}$$so you are looking for a solution of your PDE like$$
F(x,t) =a_0 + \sum_{n=0}^\infty \phi_n(x)G_n(t)=
a_0 + \sum_{n=0}^\infty \left( A_n \sin (2n \pi x) + B_n \cos (2n \pi x) \right)
e^{4n^2\pi^2 t}$$It is only now that you can use the condition $$F(x,0)=x^2$$Also I have included a constant term ##a_0##. You need to work the case ##\lambda=0## to verify you get a constant term.