6.1.5 problem number 5

problem number 421

Added January 2, 2019.

Problem 1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica

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

Maple

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

Hand solution

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

We can treat this similar to linear ODE and use an integrating factor \(I=e^{-\int f\left ( x,y\right ) dx}\) hence the above becomes\begin{align*} \frac{\partial }{\partial x}\left ( we^{-\int f\left ( x,y\right ) dx}\right ) & =e^{-\int wdx}g\left ( x,y\right ) \\ we^{-\int f\left ( x,y\right ) dx} & =\int e^{-\int f\left ( x,y\right ) dx}g\left ( x,y\right ) dx+G\left ( y\right ) \\ w & =\left ( \int e^{-\int f\left ( x,y\right ) dx}g\left ( x,y\right ) dx+G\left ( y\right ) \right ) e^{\int f\left ( x,y\right ) dx} \end{align*}

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