Solutions of an ODE

Question

Find all solutions of $$(at+bx+c)x'+(pt+qx+r)=0$$ Where $x$ is a function of $t$. Do it by a change of variables. I managed to reduce the problem to a equation of the type $xx'+mx+nt=0$ but I don't know if I can solve it.

Answer 1

Suppose that $aq-bp\neq0$, namely $$ at+bx+c=0,pt+qx+r=0$$ have the solution $t=t_0,x=x_0$. Let $s=t-t_0,y=x-x_0$ and the DE becomes $$ y'=-\frac{ps+qy}{as+by} $$ which is homogeneous and can be solved easily. If $$ \frac{a}{p}=\frac{b}{q}\neq \frac{c}{r}$$ or $$ \frac{a}{p}=\frac{b}{q}= \frac{c}{r}$$ It is easy to handle. You can handle yourself.