I am using Evans - Partial Differential Equations to draw inspiration for solving the biharmonic homogeneous equations in $\mathbb{R}^n$ where $n = 2$.
$$ \Delta^2 u = 0 $$
My attempt is to write the following equivalent system and solve one equation at the time
$$ \left\{ \begin{array}{l} \Delta u = v \\ \Delta v = 0 \end{array} \right. $$
The solution for $\Delta v = 0$ I refer to Evans which yields
$$ \Delta v = 0 \iff v = A\log r + B $$
where $r = \sqrt{x^2 + y^2}$. My contribution here is then to solve
$$ \Delta u= A\log r + B $$
With the change of variables above I can write
$$ \Delta = \partial_x^2 + \partial_y^2 = \left(\frac{1}{r} + \frac{d}{dr}\right) \frac{d}{dr} $$
Therefore I can equivalently solve
$$ \left(\frac{1}{r} + \frac{d}{dr}\right) \frac{d}{dr}u= A\log r + B. $$
Again taking inspiration from Evans I get the following system
$$ \left\{ \begin{array}{l} \left(\frac{1}{r} + \frac{d}{dr}\right) w = A\log r + B \\ \frac{d}{dr} u = w \end{array} \right. $$
I solve the first equation of this last system using integrating factor
$$ \left(\frac{1}{r} + \frac{d}{dr}\right) w = A\log r + B \iff \left(1 + r \frac{d}{dr}\right) w = r \left(A\log r + B\right) \iff \frac{d}{dr} \left( rw \right) = r \left(A\log r + B\right) \iff rw = \frac{r^2}{2} \left(A\log r + B\right) - \frac{A}{4}r^2 + C \iff w = \frac{r}{2} \left(A\log r + B\right) - \frac{A}{4}r + \frac{C}{r}. $$
Now I can solve the second equation
$$ w = \frac{d}{dr} u = \frac{r}{2} \left(A\log r + B\right) - \frac{A}{4}r + \frac{C}{r} $$
where I integrate side by side and this gives me
$$ u = \frac{r^2}{4} \left(A\log r + B\right) - \frac{A}{8} r^2 - \frac{A}{8}r^2 + C \log r + D = \frac{r^2}{4} \left(A\log r + B\right) - \frac{A}{4} r^2 + C \log r + D. $$
Rearranging a bit I get
$$ u = \left(C + \frac{A}{4}r^2 \right) \log r + \left(B - A\right)\frac{r^2}{4} + D. $$
Question: Is this solution correct? also the procedure being used is it the standard one?
Here's a reference to an old book with the similar solution to the biharmonic equation. I've found it through some later articles referring back to it, so it is perhaps one of a standard references:
Greenberg. Applications of Green's Functions in Science and Engineering 1971, 2015
p. 125 onwards
you can download a djvu scan here: https://ebin.pub/applications-of-greens-functions-in-science-and-engineering-0486797961.html