# How would you solve the next ODE? (dy)/(dt) =(at + by + m)/( ct + dy + n), where a,b,c,d,m,n are constants and ad=bc.

Separable Differential Equation
How would you solve the next ODE?
$\frac{dy}{dt}=\frac{at+by+m}{ct+dy+n},$
where $a,b,c,d,m,n$ are constants and $ad=bc$
Corrected.
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Taniyah Lin
Let $u=at+by$ then $ct+dy=Ku$ for some constant $K$ (as $ad=bc$). Now $\frac{du}{dt}=a+b\frac{dy}{dt}$ and $\frac{dy}{dt}=-\frac{a}{b}+\frac{1}{b}\frac{du}{dt}$. Next we have $\frac{dy}{dt}=\frac{u+m}{Ku+n}$ so $-\frac{a}{b}+\frac{1}{b}\frac{du}{dt}=\frac{u+m}{Ku+n}$. And this can be solved by separation as it is autonomous.