Solve the differential equation by separation of variables: $\frac{dp}{dt}=p-p^2$. Which of the following is the general solution?

• A. $p=\frac{1}{1+C{e}^{-t}}$
• B. $p=C{e}^{-t}$
• C. $p=C{e}^t$
• D. $p=\frac{1}{C{e}^t-1}$

Answer: (A)

Verified step by step guidance

1. Step 1: Start by rewriting the differential equation dp/dt = p - p^2 in a form suitable for separation of variables. Factorize the right-hand side to get dp/dt = p(1 - p).
2. Step 2: Separate the variables p and t. Divide both sides by p(1 - p) and multiply through by dt to obtain (1 / (p(1 - p))) dp = dt.
3. Step 3: Break down the left-hand side into partial fractions. Express 1 / (p(1 - p)) as A/p + B/(1 - p), where A and B are constants to be determined. Solve for A and B by equating coefficients.
4. Step 4: Integrate both sides. For the left-hand side, integrate the partial fractions term by term: ∫(A/p) dp + ∫(B/(1 - p)) dp. For the right-hand side, integrate ∫dt. This will yield logarithmic expressions for p and a linear expression for t.
5. Step 5: Solve for p explicitly. Combine the constants of integration and simplify the expression to match the form of the general solution. Use algebraic manipulation to arrive at p = 1 / (1 + Ce^{-t}), where C is the constant of integration.