Solution of a differential equation in space of distributions

Question

I cant figure out how to find general solution to equation $(1+x)^2 u''=0$ in the space of distributions. Any ideas?

Answer 1

Obviously, supp $u'' \subseteq \{-1\}$, so $ u''$ is linear combination of derivatives of $\delta_{-1}$. As the general solution of $x^2 S = 0$ is $S=c_1\delta + c_2\delta'$, this implies $$ u'' = c_1\delta_{-1} + c_2\delta_{-1}'. $$ and hence $$ u(x)=c_1(1+x)H(1+x) + c_2H(1+x) + c_3x + c_4 $$ where $H$ denotes the Heaviside function.