In this problem, y = 1/(x2 + c. is a one-parameter family of solutions of the first-order de y' + 2xy2 = 0. find a solution of the first-order ivp consisting of this differential equation and the given initial condition. y(4) = 1/15

A family of solutions to the DE is given to be y = 1/(x^2 + c);
With given initial condition y(4) = 1/15.

Let's use this initial data to find a particular solution.
Plug 4 in for x, and 1/15 in for y,

1/15 = 1/(4^2 + c)
Solve for c.

Reciprocate each side,
15 = 4^2 + c
-1 = c

Plugging this c value back into our family of solutions will give us one particular solution to the DE,

y = 1/(x^2 - 1)