Question

(4𝑥𝑦 + 3𝑦2 )𝑑𝑥 + 𝑥(𝑥 + 2𝑦)𝑑𝑦 = 0

Answer 1

$(4xy+3y^2)_y=4x+6y\neq (x(x+2y))_x=2x+2y$

So, the equation is not exact.

$u=y/x$

$y'=xu'+u$

$3x^2u^2+2x^2(xu)'+4x^2u+x^2(xu)'=0$

$2x^3uu'+x^3u'+5x^2u^2+5x^2u=0$

$\frac{(2u+1)u'}{5u(u+1)}=-\frac{1}{x}$

$\int\frac{2u+1}{5u(u+1)}du=-\int\frac{1}{x}dx$

$\frac{ln(u^2+u)}{5}=c-lnx$

$u_1=-\frac{\sqrt{c_1/x+1}+1}{2}$

$u_2=\frac{\sqrt{c_1/x+1}-1}{2}$

$y_1=-\frac{\sqrt{c_1/x+1}+1}{2}x$

$y_2=\frac{\sqrt{c_1/x+1}-1}{2}x$