In this problem, y = 1 / (1 + c1e-x) is a one-parameter family of solutions of the first-order de y' = y - y2. find a solution of the first-order ivp consisting of this differential equation and the given initial condition. y (0) = - 1 8

The solution can be determined if we can specify the value of the arbitrary constant C₁. We do this by using the initial condition. When x=0, y is equal to - 18. Substitute this to the general solution.

y = 1 / (1 + C₁e⁻ˣ)

-18 = 1 / (1 + C₁e⁰)

-18 = 1 / (1+C₁)

C₁ = - 1 - 1/18 = - 19/18

Therefore, the specific solution is:

y = 1 / (1 - 19e⁻ˣ/18)