Answer to Question #153386 in Differential Equations for Robin

Question

Complete integral of pq=xy

Answer 1

Given ,

$\;\;\;\;pq=xy\\ \Rightarrow \frac{p}{x}=\frac{y}{q}$

Let

$\;\;\;\;\frac{p}{x}=\frac{y}{q}=k\\ \Rightarrow p=xk\;\;\;and, \;\;\;q=\frac{y}{k}$

Now,

$dz=pdx+qdy=xkdx+\frac{y}{k}dy$

Integrating both sides,

$\;\;\;\int dz=k\int xdx+\frac{1}{k}\int ydy\\\;\\ \Rightarrow z=\frac{kx^2}{2}+\frac{y^2}{2k}+c' \\ [c'\;integrating\;constant]\\\;\\ \Rightarrow2kz=k^2x^2+y^2+c\\ [c=2c'k]$

$\therefore$ So, the required equation :-

$$\boxed{2kz=k^2x^2+y^2+c}$$