6.1.6 problem number 6

problem number 421

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


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