I have this problem:
Let $f \in C_{o}^{\infty} (\mathbb{R}^n)$. Propose formulas of the form $u(x) = \displaystyle\int_{}^{} \frac{\widehat{f}(\delta) e^{ix\delta}}{p(\delta)} d \delta$ to solve:
i) $\displaystyle\sum_{j=1}^{n} \frac{\partial^4 u(x)}{\partial x_{j}^4} + u(x) = f(x)$
and
ii) $\displaystyle\sum_{k,l}^{} \frac{\partial^4 u(x)}{\partial x_{k}^2 \partial x_{l}^2} - 2 \cdot \displaystyle\sum_{k=1}^{n} \frac{\partial^2 u(x)}{\partial x_{k}^2} + u(x) = f(x)$.
I honestly don't know where to start, I don't know what I should use. If someone can give me some help as to how I should start solving, I am very grateful. (I believe that $\hat{f}$ is the Fourier transform. I say "I believe" because it is not specified in the statement).
For a general linear differential operator $L[u]$ with constant coefficients you know that to get a particular solution to $$ L[u]=ce^{iδ·x} $$ you make a trial approach with $u=Ae^{iδ·x}$ and you get $L[e^{iδ·x}]=p(δ)e^{iδ·x}$. So $A=\frac{c}{p(δ)}$. If the right side is a sum of exponential terms, $$ L[u]=\sum_{k=1^n}c_ke^{iδ_k·x} $$ then the particular solution is composed of the particular solutions for each term, $$ u_p(x)=\sum_{k=1^n}\frac{c_k}{p(δ_k)}e^{iδ_k·x}. $$ Now go to the limit in some fashion, for instance via step functions, to apply an analogous argument when the right side is a function with a Fourier transform, $$ L[u](x)=f(x)=\int_{D}\hat f(δ)e^{iδ·x}\,dδ. $$