6.1.3 problem number 3

problem number 418

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 ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] == w[x, y]*f[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := diff(w(x,y),x)=w(x,y)*f(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (y \right ) {\mathrm e}^{\int f \left (x , y\right )d x}\]

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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