Question

Solve:
$\frac{dy}{dx}=\frac{3x-2y-5}{3y-2x+5}$

Answer 1

$\frac{dy}{dx}=\frac{3x-2y-5}{3y-2x+5}$ .........$\left(1\right)$

Let $X=x+a$

$Y=y+b$

$\frac{dY}{dX}=\frac{3X-2Y}{2Y-2X}$

On putting $\to X$ & $Y$

$\frac{dy}{dx}=\frac{3x-2y+3a-2b}{3y-2x+3b-2a}$

On comparing above equation with equation $1$

$3a-2b=-5$ ........$\left(2\right)$

$3b-2a=5$ .......$\left(3\right)$

equation $\left(2\right)$ & equation $\left(3\right)$

$a+b=0$

$a=-b$

Putting equation $\left(2\right)$

$=5b=-5$

$b=1$ & $a=-1$

$X=x-1$

$Y=y+1$

Reduced homogeneous equation is

$\frac{dY}{dX}=\frac{3X-2Y}{3Y-2X}$

$\frac{dY}{dX}=\frac{3-2Y/X}{3Y/X-2}$ .....$\left(4\right)$

Let $\frac{Y}{X}=v$

$\frac{dY}{dX}=v+X\frac{dv}{dX}$

Substituting in equation $\left(4\right)$