2.1.49 \(u_x+x u_y=(y-\frac {1}{2}x^2)^2\) with \(u(0,y)=e^{y}\). Problem 3.5(f) Lokenath Debnath

problem number 49

Added June 3, 2019.

Problem 3.5(f) nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for \(u(x,y)\) \[ u_x+x u_y=(y-\frac {x^2}{2})^2 \] with \(u(0,y)=e^{y}\).

Mathematica


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

Maple


\[u \left (x , y\right ) = \frac {x^{5}}{4}-x^{3} y +x y^{2}+{\mathrm e}^{-\frac {x^{2}}{2}+y}\]

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