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Solve the following initial value problem. y'+2x(y + 1) = 0, y(0) = 2

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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)

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