The given differential equation is:

\[
y' = 4y - y^2
\]

This is a first-order nonlinear ordinary differential equation. To solve it, we can use the method of separation of variables. Let's go through the steps:

1. **Rewrite the equation** to separate variables \( y \) and \( x \):

\[
\frac{dy}{dx} = 4y - y^2
\]

We can factor the right side:

\[
\frac{dy}{dx} = y(4 - y)
\]

2. **Separate the variables** \( y \) and \( x \):

\[
\frac{dy}{y(4 - y)} = dx
\]

3. **Integrate both sides**:

For the left side, we need to decompose the fraction using partial fractions:

\[
\frac{1}{y(4 - y)} = \frac{A}{y} + \frac{B}{4 - y}
\]

Multiplying both sides by \( y(4 - y) \) gives:

\[
1 = A(4 - y) + By
\]

To find \( A \) and \( B \), equate the coefficients:

- \( A(4) = 4A \), so \( A = \frac{1}{4} \).
- \( B = \frac{1}{4} \).

Thus, we have:

\[
\frac{1}{y(4 - y)} = \frac{1/4}{y} + \frac{1/4}{4 - y}
\]

So the integral becomes:

\[
\int \left( \frac{1/4}{y} + \frac{1/4}{4 - y} \right) dy = \int dx
\]

This simplifies to:

\[
\frac{1}{4} \ln |y| - \frac{1}{4} \ln |4 - y| = x + C
\]

4. **Simplify the expression**:

Combine the logarithms:

\[
\ln \left|\frac{y}{4 - y}\right| = 4x + C_1
\]

Exponentiating both sides:

\[
\frac{y}{4 - y} = Ce^{4x}
\]

Where \( C = e^{C_1} \) is a constant.

5. **Solve for \( y \)**:

\[
y = \frac{4Ce^{4x}}{1 + Ce^{4x}}
\]

This is the general solution to the differential equation.

Would you like more details on this process or have any questions?

### Related Questions:
1. How do you solve separable differential equations?
2. What are the conditions for the existence of a unique solution to a first-order ODE?
3. How can you find a particular solution given an initial condition?
4. What are the qualitative behaviors of solutions to logistic equations?
5. How does the method of partial fractions work in integrals?

**Tip:** When solving nonlinear differential equations, separating variables and integrating is a common technique, but always check if the equation can be factored or simplified first.

Explore the detailed step-by-step solution for solving the nonlinear differential equation y' = 4y - y^2 using separation of variables and partial fractions. This guide covers key mathematical concepts and provides a comprehensive explanation suitable for students in Grades 11-12 or college level.

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