Answer to Question #168144 in Differential Equations for Abhishek raj

Question

2p+3q=1

Answer 1

 The given partial differential equation can be written as $Pp+Qq=R$ where $P=2, Q=3,$ and $R=1.$ The 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.