Final answer:
To solve the given initial value problem, we can use the method of variable separation. First, rewrite the equation as dy/dx = -2x(y + 1). Then, separate the variables and integrate both sides. Solve for the constant of integration using the initial condition and substitute it back into the equation.
Step-by-step explanation:
To solve the given initial value problem, we can use the method of variable separation. First, let's rewrite the equation as: dy/dx = -2x(y + 1). Now, we can separate the variables and integrate both sides:
1/(y + 1) dy = -2x dx
Integrating both sides will give us the solution:
ln|y + 1| = -x^2 + C
We can solve for the constant of integration, C, using the initial condition y(0) = 2. Plugging in the values, we get:
ln|2 + 1| = -0^2 + C
Simplifying, we find: C = ln(3). Finally, substituting this value back into the equation, we have:
ln|y + 1| = -x^2 + ln(3)