# Thread: General Solution to a Differential Equation

1. ## General Solution to a Differential Equation

Find the general solution to x^2y' + y^2(x^2+1)=0.

2. ## Separable

This is a first order nonlinear differential equation, so there will be one arbitrary constant in the solution, and we cannot use the standard methods that we would for a linear equation. However, this one is separable:
$\displaystyle x^2 y' + y^2 (x^2 + 1) = 0$
$\displaystyle x^2 y' = -y^2 (x^2 + 1)$

$\displaystyle \dfrac{1}{y^2}\dfrac{dy}{dx} = -\dfrac{x^2 + 1}{x^2}$

$\displaystyle \dfrac{dy}{y^2} = -(1+\dfrac{1}{x^2})dx$

From here you can integrate, and solve the result for y to get the general equation.

--Kevin C.