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The given differential equation is

\[ \frac{dy}{dt} = 0.7944 y \left(1 - \frac{y}{64}\right), \quad y(0) = 2. \]

This is a separable differential equation. To solve it, we'll first separate the variables and then integrate.

### Step-by-Step Solution

1. **Separate the variables:**
\[ \frac{dy}{y \left(1 - \frac{y}{64}\right)} = 0.7944 \, dt. \]

2. **Simplify the left-hand side:**
\[ \frac{1}{y \left(1 - \frac{y}{64}\right)} = \frac{1}{y} + \frac{1}{64 - y}. \]

Therefore,
\[ \frac{1}{y} + \frac{1}{64 - y} \, dy = 0.7944 \, dt. \]

3. **Integrate both sides:**
\[ \int \left( \frac{1}{y} + \frac{1}{64 - y} \right) dy = \int 0.7944 \, dt. \]

\[ \int \frac{1}{y} \, dy + \int \frac{1}{64 - y} \, dy = 0.7944 \int dt. \]

4. **Solve the integrals:**
\[ \ln|y| - \ln|64 - y| = 0.7944t + C, \]

where \(C\) is the constant of integration.

5. **Combine the logarithms:**
\[ \ln \left| \frac{y}{64 - y} \right| = 0.7944t + C. \]

6. **Exponentiate both sides to solve for \(y\):**
\[ \left| \frac{y}{64 - y} \right| = e^{0.7944t + C} = Ae^{0.7944t}, \]

where \(A = e^C\) is a new constant.

7. **Solve for \(y\):**
\[ \frac{y}{64 - y} = Ae^{0.7944t}. \]

\[ y = Ae^{0.7944t} (64 - y). \]

\[ y + Ae^{0.7944t} y = 64Ae^{0.7944t}. \]

\[ y (1 + Ae^{0.7944t}) = 64Ae^{0.7944t}. \]

\[ y = \frac{64Ae^{0.7944t}}{1 + Ae^{0.7944t}}. \]

8. **Determine the constant \(A\) using the initial condition \(y(0) = 2\):**

When \(t = 0\),
\[ 2 = \frac{64A}{1 + A}. \]

\[ 2 (1 + A) = 64A. \]

\[ 2 + 2A = 64A. \]

\[ 2 = 62A. \]

\[ A = \frac{1}{31}. \]

9. **Substitute \(A\) back into the solution:**
\[ y = \frac{64 \left(\frac{1}{31}\right)e^{0.7944t}}{1 + \left(\frac{1}{31}\right)e^{0.7944t}}. \]

\[ y = \frac{\frac{64}{31} e^{0.7944t}}{1 + \frac{1}{31} e^{0.7944t}}. \]

\[ y = \frac{\frac{64}{31} e^{0.7944t}}{\frac{31}{31} + \frac{1}{31} e^{0.7944t}}. \]

\[ y = \frac{64 e^{0.7944t}}{31 + e^{0.7944t}}. \]

### Final Answer:
\[ y(t) = \frac{64 e^{0.7944t}}{31 + e^{0.7944t}}. \]

Would you like more details or have any questions about this solution?

Here are 5 questions you can ask next:
1. How do you solve differential equations using the method of integrating factors?
2. What is the Laplace transform and how is it used to solve differential equations?
3. Can you explain the method of undetermined coefficients for solving non-homogeneous differential equations?
4. How do you solve a system of linear differential equations?
5. What are some applications of differential equations in real-world problems?

**Tip:** When solving differential equations, always verify your solution by substituting it back into the original equation to ensure it satisfies both the equation and the initial conditions.

Learn how to solve the differential equation dy/dt = 0.7944 y (1 - y/64) with detailed step-by-step instructions. This problem involves solving separable differential equations using integration methods. Suitable for undergraduate-level students and professionals.

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