Final answer:
To solve the Bernoulli differential equation (dy/dx) - 2y = y², we can use the substitution method. The solution to the equation is y = (4e^(2x) - 1)/3.
Step-by-step explanation:
To solve the Bernoulli differential equation (dy/dx) - 2y = y², we can use the substitution method. Let's substitute y = u^-1 into the equation and solve for u. First, differentiate u^-1 with respect to x: du/dx = -u^-2 * du/dx. Substituting this into the equation, we get -u^-2 * du/dx - 2 * (u^-1) = (u^-1)^2.
Simplifying the equation, we get -du/dx - 2u = u^2. This is now a linear differential equation, which can be solved using standard techniques. Rearranging the equation, we get du/dx + 2u = -u^2.
Using an integrating factor, we multiply both sides of the equation by e^(2x). This gives us e^(2x) * du/dx + 2e^(2x) * u = -u^2 * e^(2x). Applying the product rule to the left side, we get (e^(2x) * u)' = -u^2 * e^(2x).
Now, we substitute back y = u^-1 into the equation. This gives us y = (Ce^(2x) - 1)/3. Applying the initial condition y(0) = 1, we get 1 = (C - 1)/3, which implies C = 4.
Therefore, the solution to the Bernoulli differential equation (dy/dx) - 2y = y², with the initial condition y(0) = 1, is y = (4e^(2x) - 1)/3.