6.1.3 problem number 3

problem number 419

Added January 2, 2019.

Problem 1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x = w f(x,y) \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1(y) \exp \left (\int _1^xf(K[1],y)dK[1]\right )\right \}\right \}\]

Maple

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

Hand solution

\begin{align*} \frac{\partial w}{\partial x} & =wf\left ( x,y\right ) \\ \frac{1}{w}\frac{\partial w}{\partial x} & =f\left ( x,y\right ) \end{align*}

Integrating both sides w.r.t. \(x\) gives\begin{align*} \ln \left ( w\right ) & =\int _{0}^{x}f\left ( s,y\right ) ds+G\left ( y\right ) \\ w & =e^{\int _{0}^{x}f\left ( s,y\right ) ds+G\left ( y\right ) }\\ & =F\left ( y\right ) e^{\int _{0}^{x}f\left ( s,y\right ) ds} \end{align*}

Where \(F\left ( y\right ) =e^{G\left ( y\right ) }\)

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