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=\frac1{x^2+C}\implies y'=-(2x)/((x^2+C)^2)`

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

`-(2x)/((x^2+C)^2)+2x\left(\frac1{x^2+C}\right)^2=0`

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