Question

Solve:

x²y²(2ydx + xdy) - (5ydx + 7xdy) =0

Answer 1

We write the equation as

$$2x^2y^3dx+x^3y^2dy-5ydx-7xdy=0$$

Let's select the total differential x^3y^3

$$\frac{1}{3} d(x^3y^3)+x^2y^3dx-5ydx-7xdy=0.$$

Let's select the total differential -5xy

$$\frac{1}{3} d(x^3y^3)-5d(xy)+x^2y^3dx-2xdy=0.$$

Consider the equation

$$x^2y^3dx-2xdy=0$$

get

$$xdx=\frac{2dy}{y^3}.$$

As result

$$\frac{x^2}{2}+\frac{1}{y^2}=C$$

where C is constant.

Hence. the general solution of the equation is

$$\frac{x^3y^3}{3}-5xy+\frac{x^2}{2}+\frac{1}{y^2}=c$$