Question

solve the follow homogenous equation of differential eq (2x-3y-1)dx (x 3y-5)dy=0

Answer 1

We have given the differential equation,

$(2x-3y-1)dx + (x -3y-5)dy = 0$

$\dfrac{dy}{dx} + \dfrac{(2x -3y -1)}{(x -3y -5)} = 0$

Putting $x = X+h$ and $y = Y+k$

$\rightarrow 2h - 3k -1 = 0\\and\text{ } h - 3k -5 = 0$

Hence, $\dfrac{dY}{dX} + \dfrac{(2X-3Y)}{(X+3Y)} = 0$

Putting $Y = vX$

$v+X\dfrac{dv}{dX} + \dfrac{2-3v}{1-3v} = 0$

$X\dfrac{dv}{dX} = \dfrac{(3v-2)}{(1-3v)} -v$

$= \dfrac{(2v-2-3v^2)}{(1-3v)}$

$\int(\dfrac{1-3v}{2v-2-3v^2})dv + \int\dfrac{dX}{X} = logC$

$\dfrac{1}{2}log(2v-2-3v^2)+logX=logc$

Hence, this is the final solution of the given differential equation.