I'm completely new to differential equations. I'm just doing random problems online about them. One simple problem asks to re-arrange the following so that the x's and y's are all on one side:
$x^2+dy/dx +xy=1 $
Can I just use algebra to do it? Like:
$dy/dx= x^2/xy $
This equation $$x^2+dy/dx +xy=1$$ is not separable so you can not rearrange it so all $x$ terms are on one side and and all $y$ terms go on the other side.
The equation is linear and we can solve it easily by multiplying both sides by integrating factor.
$$dy/dx +xy=1-x^2$$ where integrating factor is $e^{x^2/2}$
$$\frac {d}{dx} (ye^{x^2/2})=(1-x^2) e^{x^2/2}$$ Which is solved by integrating both sides.
