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)