2.16.10 Using integral transforms.

problem number 149

Added Oct 6, 2019.

Taken from https://www.mapleprimes.com/posts/211274-Integral-Transforms-revamped-And-PDE

Solve \[ {x}^{2}{\frac {\partial ^{2}}{\partial {x}^{2}}}u \left ( x,y \right ) + x{\frac {\partial }{\partial x}}u \left ( x,y \right ) +{\frac { \partial ^{2}}{\partial {y}^{2}}}u \left ( x,y \right ) =0 \] With boundary conditions \begin {align*} u \left ( x,1 \right ) &=\begin {cases} 1& 0\leq x \hspace {5 pt} \text {and} \hspace {5pt} x<1\\ 0&1<x\\ \end {cases}\\ u(x, 0) &= 0 \end {align*}

Mathematica


\[\left \{\left \{u(x,y)\to \frac {\text {Integrate}\left [\frac {x^{-\frac {149}{33}-i K[1]} \csc \left (\frac {149}{33}+i K[1]\right ) \sin \left (y \left (\frac {149}{33}+i K[1]\right )\right )}{\frac {149}{33}+i K[1]},\{K[1],-\infty ,\infty \},\text {Assumptions}\to K[1]\in \mathbb {R}\right ]}{2 \pi }\right \}\right \}\]

Maple


\[u \left (x , y\right ) = \]