Question

In this problem,y = 1/(x2 + c)is a one-parameter family of solutions of the first-order DEy' + 2xy2 = 0.Find a solution of the first-order IVP consisting of this differential equation and the given initial condition.

Answer 1

Differentiate the given solution:

$y=\dfrac1{x^2+C}\implies y'=-\dfrac{2x}{(x^2+C)^2}$

Substitute $y$ and $y'$ into the ODE:

$-\dfrac{2x}{(x^2+C)^2}+2x\left(\dfrac1{x^2+C}\right)^2=0$

and it's easy to see the left side indeed reduces to 0.