Answer to Question #164341 in Differential Equations for murtaza irfan

Question

Q2. (c) Find the general solution of the first-order linear partial differential equation xux + yuy = 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-

$\text{lnx}=\text{lny}+C_1\\ \text{lnx}-\text{lny}=\text{ln}e^{C1}$

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

Now, taking last two terms

$\Rightarrow\dfrac{dy}{y}=\dfrac{du}{u}\\\Rightarrow lny=lnu+C_2\\ \Rightarrow lny=lnu+lne^{C_2}\\\\ \Rightarrow ln(\dfrac{y}{u})=lne^{C_2}$

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

$\Rightarrow \dfrac{y}{u}=C_2'$

$\Rightarrow\epsilon=\dfrac{y}{u}=C_2'$

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