Question

Find the general solution of the first-order linear partial differential equation

xu_x yu_y = u and investigate first-order, quasi-linear and linear partial differential equations

Answer 1

We have the linear differential equation

$xu_x+yu_y=u$

The system of eqautions is,

$\dfrac{dx}{x}=\dfrac{dy}{y}=\dfrac{du}{u}$

Now, taking first two terms

$\dfrac{dx}{x}=\dfrac{dy}{y}$

Integrating Both the sides-

$lnx=lny+C1\\ lnx-lny=\text{ln}e^{C1}$

$(\dfrac{x}{y})=e^{C1}\\ \dfrac{x}{y}=C1'\\ \phi=\dfrac{x}{y}=C1'$

Now, taking last two terms

$\dfrac{dy}{y}=\dfrac{du}{u}\\ lny=lnu+C2\\ lny=lnu+lne^{C2}\\\\ ln(\dfrac{y}{u})=lne^{C2}$

$\dfrac{y}{u}=e^{C2}$

$\dfrac{y}{u}=C2'$

$\epsilon=\dfrac{y}{u}=C2'$

Hence, the general solution is $f(\phi,\epsilon)=f(\dfrac{x}{y},\dfrac{y}{u})=0$