This photo is a set of exercises from a book on Fourier Analysis on the very first chapter. These exercises seem a very tedious calculation. I expected the heat equation and also Laplace's equation to be solved using separation of variables in which case exponential, sine and cossine terms would appear.
How would I deduce these other seemingly more arbitrary solutions from scratch? I would like to start with the differential equations and find a solution $u$ (not the other way around).
EDIT: It seems Laplace's equation can be solved with the assumption of a radial solution: $u=f(r)$ where $r=\sqrt{\sum_i x^2_i}$. In this case:
$$\frac{\partial f(r)}{\partial x_i}=f'(r) \frac{1}{2r}2x_i=f'(r)\frac{x_i}{r}$$ $$f_{x_i^2}(r)=f''(r)\frac{x_i^2}{r^2}+f'(r)\left(\frac{1}{r}+x_i\frac{-1}{r^2}\frac{1}{2r}2x_i\right)$$
$$\nabla^2 u=0\Rightarrow \sum_i\left( f''(r)\frac{x_i^2}{r^2}+f'(r)\left(\frac{1}{r}+x_i\frac{-1}{r^2}\frac{1}{2r}2x_i\right)\right)=f''(r) +f'(r)\left(\frac{n}{r}-\frac{1}{r}\right)=0 $$
$$ \frac{f''(r)}{f'(r)}=\frac{1-n}{r}\Rightarrow \ln(f'(r))=\ln(r^{1-n})+\tilde{C}\Rightarrow f'(r)=Cr^{1-n} $$
This really has different solutions for $n=2$ and $n=3$, namely:
$$ f=\begin{cases} C\ln(r)+D \:,\: n=2\\ C\frac{1}{r}+E \: ,\: n=3\end{cases}$$
Therefore I only really want some motivation for the solution to the heat equation (that is, problems 1 and 2).
