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We are given a differential equation of the form:

\[
y' = \frac{1}{40} \ln\left(\frac{2}{3}\right)(y - 20)
\]

We are tasked with finding the general solution for \( y(t) \), in the form:

\[
y(t) = A + Be^{Ct}
\]

This is a first-order linear differential equation that can be solved using separation of variables or recognizing it as a form of an exponential growth/decay equation.

### Step-by-step solution:

1. **Rewrite the equation**: The given equation is already in a separable form:
   \[
   \frac{dy}{dt} = \frac{1}{40} \ln\left(\frac{2}{3}\right)(y - 20)
   \]
   
2. **Separate variables**: Move all \( y \)-related terms to one side and \( t \)-related terms to the other:
   \[
   \frac{1}{y - 20} dy = \frac{1}{40} \ln\left(\frac{2}{3}\right) dt
   \]

3. **Integrate both sides**:
   - Integrating the left-hand side:
     \[
     \int \frac{1}{y - 20} dy = \ln|y - 20|
     \]
   - Integrating the right-hand side:
     \[
     \int \frac{1}{40} \ln\left(\frac{2}{3}\right) dt = \frac{1}{40} \ln\left(\frac{2}{3}\right) t + C
     \]
     where \( C \) is the constant of integration.

4. **Solve for \( y \)**:
   \[
   \ln|y - 20| = \frac{1}{40} \ln\left(\frac{2}{3}\right) t + C
   \]
   Exponentiating both sides to remove the logarithm:
   \[
   |y - 20| = e^{\frac{1}{40} \ln\left(\frac{2}{3}\right) t + C}
   \]
   Simplifying:
   \[
   y - 20 = e^{C} \cdot e^{\frac{1}{40} \ln\left(\frac{2}{3}\right) t}
   \]
   Let \( B = e^{C} \), then:
   \[
   y - 20 = B \cdot e^{\frac{1}{40} \ln\left(\frac{2}{3}\right) t}
   \]
   
5. **Final solution**: Solving for \( y \), we get:
   \[
   y(t) = 20 + B \cdot e^{\frac{1}{40} \ln\left(\frac{2}{3}\right) t}
   \]

This is in the required form \( y(t) = A + Be^{Ct} \), where:
- \( A = 20 \),
- \( B \) is a constant determined by initial conditions,
- \( C = \frac{1}{40} \ln\left(\frac{2}{3}\right) \).

---

Let me know if you'd like further details or explanations!

Here are 5 related questions:
1. How can we apply initial conditions to find the constant \( B \)?
2. What does the constant \( C \) signify in this differential equation?
3. How would the solution change if the logarithmic term had a different base?
4. Can we determine the long-term behavior of \( y(t) \) as \( t \to \infty \)?
5. How do we interpret this equation in the context of population growth/decay?

**Tip:** When solving differential equations, always check if the equation is separable or linear. This can simplify the integration process significantly!

y' = (1/40) ln(2/3) (y - 20). Find the formula for y(t) in the form A + Be^(Ct), where A, B, and C are constants to be determined.

Learn how to solve the differential equation y' = (1/40) ln(2/3) (y - 20) to find the formula for y(t) in the form A + Be^(Ct). This involves concepts like separation of variables and exponential functions. Suitable for undergraduate students.

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