2.1.62 \(2 x u_x+(x+1) u_y=y\) with \(u(1,y)=2 y\). Problem 3.14(d) Lokenath Debnath

problem number 62

Added June 3, 2019.

Problem 3.14(d) nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for \(u(x,y)\) \[ 2 x u_x+(x+1) u_y=y \] With \(u(1,y)=2 y\) with \(x>0\)

Mathematica


\[\left \{\left \{u(x,y)\to \frac {1}{8} \left (-2 (x-2 y+4) \log (x)-6 x-\log ^2(x)+16 y+6\right )\right \}\right \}\]

Maple


\[u \left (x , y\right ) = -\frac {\ln (x )^{2}}{8}-\frac {3 x}{4}+2 y +\frac {\left (-2 x +4 y -8\right ) \ln (x )}{8}+\frac {3}{4}\]

________________________________________________________________________________________