Answer to Question #233801 in Differential Equations for Phyroehan

Question

Find the general/particular solution of the following Differential Equations.

(Integrable Combinations)

y(y² + 1)dx + x(y² - 1)dy=0

Answer 1

Solution

Let’s rewrite eq. in the form

$y^3dx+xy^2dy+ydx-xdy=0$

Dividing by y² we obtain

$ydx+xdy+\frac{ydx-xdy}{y^2}=0$

According to Product Rule and Quotient Rule

$d\left(xy\right)+d\left(\frac{x}{y}\right)=0$

Thus integrating

$xy+\frac{x}{y}=C$

Here C is arbitrary constant.

Answer: xy+x/y=C