I have the PDE $$\nabla^4\phi + 2B\nabla^2\phi + C\phi = 0$$ where $B$ and $C$ are two reals numbers and $\phi$ depends on space variables. I am looking for solutions under the form $x\mapsto e^{ikx}$. It gives a new equation $$k^4-2Bk^2+C=0$$ We found two real numbers distinct and positive: $k_1^2=B+\sqrt{B^2-C}$ and $k_2^2=B-\sqrt{B^2-C}$.
Then is it possible to split the first equation into $$\nabla^2\phi_1 + k_1^2 \phi_1 = 0$$ $$\nabla^2\phi_2 + k_2^2 \phi_2 = 0$$ with $\phi = \phi_1+\phi_2$?