2.1.63 \(x u_x+y u_y=x^2+y^2\) with \(u(x,1)=x^2\). Problem 3.14(e) Lokenath Debnath

problem number 63

Added June 3, 2019.

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

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

Mathematica


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

Maple


\[u \left (x , y\right ) = \frac {x^{2}}{2}+\frac {y^{2}}{2}+\frac {x^{2}}{2 y^{2}}-\frac {1}{2}\]

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