Solve the following differential equation:
$a(x\frac{dy}{dx}+2y)=xy\frac{dy}{dx}$. –Edited: see edit notes
I am having trouble solving this equation, problems that I run into are outlined below.
First, this is a non-exact differential equation. I will not put the work here, but it can be seen if you put the equation in the form $\frac{dM}{dy}=\frac{dN}{dx}$, and it comes out that $\frac{dM}{dy}\neq\frac{dN}{dx}$. So, integrating factors must be used in order to move forward solving.
But this becomes very difficult. If you choose to multiply your ODE by some function $\mu(x)$ or a $\mu(y)$ or $\mu(x,y)$, attempting to find the integrating factor, nothing cancels and you are left integrating something that cannot be integrated.
I also tried to multiply the ODE by $x^\alpha y^\beta$ (because I originally thought that $N$ and $M$ were sums of products of powers of $x$ and $y$, but this also proved inadequate, as the $a$s in the ODE did not cancel, and you are left with two sides of an equation that you cannot get to equal one another.
Leaving me back where I started- ground zero. Does anyone have any ideas on how to find this integrating factor?
Best Answer
The equation is separable,
$$2a^2y=x(y-a)\frac{dy}{dx},$$
$$2a^2\frac{dx}{x}=\frac{y-a}{y}dy,$$
$$2a^2\log x+C=y-a\log y.$$