I am interested in this function
$\Phi=(C_1Ai(\frac{-A a x_2-(\nu a^2-\nu c^2+d)}{\nu (\frac{A a}{\nu})^{\frac{1}{3}}})+C_2Bi(\frac{-A a x_2-(\nu a^2-\nu c^2+d)}{\nu (\frac{A a}{\nu})^{\frac{1}{3}}}))(C_3 sin(c x_3)+C_4 cos(c x_3))e^{a x_1}e^{-dT}$
This equals:
$\Phi=-\frac{\partial^2 u_2'}{\partial x_1^2}-\frac{\partial^2 u_2'}{\partial x_2^2}-\frac{\partial^2 u_2'}{\partial x_3^2}$
I do not know how to solve this for $u_2'$. Usually I would use Greens functions to solve the Poisson equation, but due to the "inhomogenity" being a function in terms of AiryAi and AiryBi functions I am not sure how to handle them. Another idea was to assume something like $u_2'=F(x_2) (C3sin(c x_3)+C4cos(c x_3))e^{a x_1}e^{-dT}$. This would lead to:
$C_1Ai(\frac{-A a x_2-(\nu a^2-\nu c^2+d)}{\nu (\frac{A a}{\nu})^{\frac{1}{3}}})+C_2Bi(\frac{-A a x_2-(\nu a^2-\nu c^2+d)}{\nu (\frac{A a}{\nu})^{\frac{1}{3}}})=-\frac{\partial^2 F(x_2)}{\partial x_2^2}+F(x_2)(c^2-a^2)$.
I would like to solve this for $F(x_2)$, but I do not know how to do it.
I have spent quite some time on this problem and I would highly appreciate any kind of help.