Question

A first order partical differential equations p+q=z-xy is

Answer 1

This is a Lagrange's equation whose general formula is pP+qQ=R

This can be solved by the formula $(dx/P)=(dy/Q)=(dz/R)$

$Here, P=1 ,Q= 1 ,R=z-xy$

$Now, {dx/1}={dy/1}={dz/z-xy}$

From the 1st two ratio,

dx= dy

=> dx-dy=0

Integrating above equation we get,

$x-y=C ―――(1)$

Now from 1st and 3rd ration,

dx=dz/z-xy

=> $x=ln(z-xy)+C$ ――――(2)

Equation (1)&(2) together yeilds the solution.

$x-x+y= ln(z-xy)$

$y= ln(z-xy)$