If we have an inhomogeneous boundary value problem $x^2 y” + xy’ + (x^2 -1)y = x,$ $y(0) = y(b) = 0,$ where $b>0$ How to use Green’s Funtion to Solve this problem. I am facing issues with equations and the number of variables. Please help me solving this

Can the behaviour $$ \text{constant}\times|x|^{2-d} $$ be obtained for the solution of the distributional equation in $\mathbb R^d$, for $d\ge 3$, $$ \Delta u(x)=\delta(x) $$ via Fourier transform method?

The fundamental solution (or Green function) for the Laplace operator in $d$ space dimensions $$\Delta u(x)=\delta(x),$$ where $\Delta \equiv \sum_{i=1}^d \partial^2_i$, is given by $$ u(x)=\begin{cases} \dfrac{1}{(2-d)\Omega_d}|x|^{2-d}\text{ for } d=1,3,4,5,\ldots\\ \dfrac{1}{2\pi}\log|x| \ \ \ \ \ \ \ \ \ \ \ \ \text{for } d=2, \end{cases} $$ where $\Omega_d$ is the $d$-dimensional solid angle […]

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