#### 6.1.3 problem number 3

problem number 419

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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