[Math] Find a one parameter family of solutions of the following first order ordinary differential equation

Question

Find a one parameter family of solutions of the following first order ordinary differential equation

$$(3x^2 + 9xy + 5y^2) dx – (6x^2 + 4xy) dy = 0$$

Hello. So I am stuck after I find out that they are not exact. Please help.

Answer 1

Fist of all, welcome to the site !

The equation is $$3 x^2+9 x y(x)+5 y(x)^2-\left(6 x^2+4 x y(x)\right) y'(x)=0$$ Looking at the last term, let $y(x)=u(x)-\frac 32 x$ to get $$-4 x u(x) u'(x)+5 u(x)^2+\frac{3 x^2}{4}=0$$ that is to say $$-2x \left(u^2(x)\right)'+5 u^2(x)+\frac{3 x^2}{4}=0$$ So, let $u(x)=\pm \sqrt{v(x)}$ to get $$-2 x v'(x)+5 v(x)+\frac{3 x^2}{4}=0$$ which looks to be simple.