Find a harmonic function with imposed restrictions

[TheCanadian]

I am trying to find a harmonic function based on the conditions imposed in the images. I see how one can make an Ansatz that $\phi(x,y) = xy + \psi(x,y)$ and can arrive at the solution given by ensuring the function satisfies the given conditions. But is there a more systematic method to solving these types of questions? In the hint, it says to use $z^2$ and it may be clear that $z^2 = x^2 + y^2 + 2ixy$, and that the imaginary part recovers a form similar to the boundary conditions, but is there something more general one can do with this hint?

Any suggestions on how to approach problems of this sort?

 

[TheCanadian]

As another example, I have attached another image. Once again, I can make the educated guess the solution has a term with $sin(4\theta)$ in the solution, but this is simply a guess I am making based on the conditions (i.e. $\phi = 0$ at $\theta = 0, \frac {\pi}{4}$). The solution says $\phi(x,y) = Im(z^4) = r^4 sin(4\theta)$. The $r^4 sin(4\theta)$ term makes sense to me and I see how it equals $Im(z^4)$, but my first guess/answer would not be something of the form $Im(z^4)$. Any idea on how to approach this problem with a better method?