6.1.6 problem number 6

problem number 422

Added January 2, 2019.

Problem 1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_y = w f(x,y)+ g(x,y) \]

Mathematica

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^yf(x,K[1])dK[1]\right ) \left (\int _1^y\exp \left (-\int _1^{K[2]}f(x,K[1])dK[1]\right ) g(x,K[2])dK[2]+c_1(x)\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) = \left ( \int \!g \left ( x,y \right ){{\rm e}^{-\int \!f \left ( x,y \right ) \,{\rm d}y}}\,{\rm d}y+{\it \_F1} \left ( x \right ) \right ){{\rm e}^{\int \!f \left ( x,y \right ) \,{\rm d}y}}\]