For Delta^2u=u_(xxxx)+2u_(xxyy)+u_(yyyy)=0 : verify a list…
For $\Delta^2u=u_{xxxx}+2u_{xxyy}+u_{yyyy}=0$: verify a list of polynomial and trig-hyperbolic solutions; and show that if $\varphi,\psi$ are harmonic then $\varphi+x\psi$, $\varphi+y\psi$, and $(x^2+y^2)\varphi$ are biharmonic.
Explanation of Method
For polynomials, apply $\Delta$ twice directly. For the trig-hyperbolic functions, compute $\Delta$ once (using $\partial_x^2\cos\lambda x=-\lambda^2\cos\lambda x$, $\partial_y^2\cosh\lambda y=\lambda^2\cosh\lambda y$) to get a simpler harmonic-looking expression, then apply $\Delta$ again. For the harmonic combinations, use $\Delta(x\psi)=x\Delta\psi+2\psi_x=2\psi_x$ (since $\Delta\psi=0$) and iterate.
Step 1 · Polynomial solutions
$\Delta(x)=0$; $\Delta(x^2)=2\Rightarrow\Delta^2(x^2)=0$; $\Delta(x^3)=6x\Rightarrow\Delta^2(x^3)=0$; $\Delta(xy)=0$; $\Delta(x^2y)=2y\Rightarrow\Delta^2=0$; $\Delta(x^3y)=6xy\Rightarrow\Delta^2=0$; $\Delta(y^4-x^4)=12(y^2-x^2)\Rightarrow\Delta^2(y^4-x^4)=24-24=0$. \checkmark (all verified symbolically).
Step 2 · Trig-hyperbolic solutions
$\Delta(x\cos\lambda x\cosh\lambda y)=-2\lambda\sin\lambda x\cosh\lambda y$, and applying $\Delta$ again to that gives $0$ (the $-\lambda^2$ from $x$ and $+\lambda^2$ from $y$ cancel). Similarly $\Delta(y\cos\lambda x\cosh\lambda y)=2\lambda\cos\lambda x\sinh\lambda y$, and $\Delta$ of that is again $0$.
Step 3 · $\varphi+x\psi$ and $\varphi+y\psi$ biharmonic
$\Delta(\varphi+x\psi)=0+x\Delta\psi+2\psi_x=2\psi_x$; applying $\Delta$ again, $\Delta(2\psi_x)=2\partial_x(\Delta\psi)=0$. Identically for $\varphi+y\psi$ via $\Delta(y\psi)=2\psi_y$.
Step 4 · $(x^2+y^2)\varphi$ biharmonic
With $r^2=x^2+y^2$: $\Delta(r^2\varphi)=\varphi\Delta(r^2)+2\nabla r^2\cdot\nabla\varphi+r^2\Delta\varphi =4\varphi+4(x\varphi_x+y\varphi_y)$. Applying $\Delta$ again: the $4\Delta\varphi$ term vanishes, and $\Delta(x\varphi_x+y\varphi_y)=2\varphi_{xx}+2\varphi_{yy}=2\Delta\varphi=0$ (using the same identity as in the previous step with $\psi=\varphi_x,\varphi_y$), so $\Delta^2(r^2\varphi)=0$.
Verification
All polynomial, trig-hyperbolic, and harmonic-combination
claims were checked symbolically and hold exactly. \checkmark
Note: the source's table of contents mislabelled this problem's entry
as "Problem 346" (a duplicate of the previous entry); the body text
itself is headed "Problem 347" and is a genuinely distinct problem, so
it is numbered 347 here.
Result
Conclusion: