Answer to Question #230767 in Differential Equations for Sameer

Question

2p+3q=1

Answer 1

The given partial differential equation can be written as $Pp+Qq=R$ where  $P=2, Q=3,$ and $R=1.$ Lagrange’s auxiliary equations are given by

$\dfrac{dx}{2}=\dfrac{dy}{3}=\dfrac{dz}{1}$

Take

$\dfrac{dx}{2}=\dfrac{dy}{3}$ $3dx=2dy$

Integrate

$3x-2y=c_1$

Take

$\dfrac{dy}{3}=\dfrac{dz}{1}$ $dy=3dz$

Integrate

$y-3z=c_2$

 The desired general solution is given by

$\phi(3x-2y, y-3z)=0$

where $\phi$ is an arbitrary function.