please help me understand the lecture note? heat equation and fourier series

Question

I don't quite understand equation 3.73 and 3.74.

To get $T(x,t)$ I thought I had to multiply F and G. How does that give equation 3.73?

I got G as e^{stuff} as in the last bit of equation 3.73. And F is as stated in the lecture note. But how does multiplying these and summing give $\frac{a_0}{2}$?

Also, why does $f(x)$ have fourier cosine expansion?

Thank you.

Answer 1

Not a very rigorous answer, but the reason for how (3.73) arises is that while the solutions you obtained of the form $T(x,t) = F(x) G(t)$ are valid, they might not satisfy your boundary conditions. However, a linear combination of solutions is also a solution to your differential equation, and so one looks for a solution of the form given by (3.73).