# Finding integrating factor for non exact differential equation

I am trying to solve non exact differential equation but I am unable to find the integration factor.

I solve the following question $$-3y\frac{dy}{dx}+2x=0$$

and I get the integrationg factor $$x^{-5/2}$$ and I use the proper way to find the integrating factor, but in our text book the integrating facot for above differential equation is finded and that is $$y/x^4$$.

• Why do you need an integrating factor, you can directly integrate to $-\frac32y^2+x^2=C$. Dec 6 '20 at 8:43
$$-3y\:dy+2x\:dx=0$$ is an EXACT differential equation.
In other words the integrating factor is $$1$$. $$d\left(-\frac32 y^2+x^2\right)=0$$ $$-\frac32 y^2+x^2=\text{constant}$$
$$-3y\frac{dy}{dx}+2x=0$$ $$-\dfrac 32 2yy'+2x=0$$ Note that $$((y^2)'=2yy'$$ So that you have: $$-\dfrac 32 (y^2)'+2x=0$$ Integrate: $$-\dfrac 32 y^2+x^2=c$$