Solve the differential equation $\frac{ds}{dr}=ks$ by separation of variables. Which of the following is the general solution?

• A. $s=C+kr$
• B. $s=C{r}^k$
• C. $s=C{e}^{-kr}$
• D. $s=C{e}^{kr}$

Answer: (D)

Verified step by step guidance

1. Step 1: Start with the given differential equation \( \frac{ds}{dr} = k s \). This equation suggests that the rate of change of \( s \) with respect to \( r \) is proportional to \( s \).
2. Step 2: Use the method of separation of variables. Rewrite the equation by dividing both sides by \( s \) (assuming \( s \neq 0 \)) and multiplying both sides by \( dr \): \( \frac{1}{s} ds = k dr \).
3. Step 3: Integrate both sides of the equation. On the left-hand side, integrate \( \int \frac{1}{s} ds \), which results in \( \ln|s| \). On the right-hand side, integrate \( \int k dr \), which results in \( k r + C_1 \), where \( C_1 \) is the constant of integration.
4. Step 4: Solve for \( s \) by exponentiating both sides to remove the natural logarithm. This gives \( s = e^{k r + C_1} \). Using properties of exponents, rewrite this as \( s = e^{C_1} e^{k r} \). Let \( C = e^{C_1} \), where \( C \) is a positive constant.
5. Step 5: Conclude that the general solution to the differential equation is \( s = C e^{k r} \). This matches the correct answer provided in the problem.