Question

Solve the equation y'=x(1+y^2) by using variable separable

Answer 1

The differential equation is

(1) $\frac {dy}{dx} =x(1+y^2).$

We saw that this is a separable equation, and can be written as

$\frac {dy}{1+y^2} =x{dx}.$

Let’s take the integrals from both parts of the equation:

$\int \frac {dy}{1+y^2} = \int x dx,$

so $\,$

$\arctan(y) = \frac {1}{2}x^2 +C, C=Const.\\$

Therefore, the general solution of a differential equation (1):

$y = \tan (\frac {1}{2}x^2 +C).$