https://r2.erweima.ai/mathbot/image/1722597936602eurkpbsg.jpg

The differential equation given in the image is:

\[
\frac{dy}{dx} = y(y + 2)
\]

This is a separable differential equation. To solve it, we can follow these steps:

### Step 1: Separate the variables

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

\[
\frac{1}{y(y + 2)} \, dy = dx
\]

### Step 2: Integrate both sides

We will integrate both sides with respect to their respective variables. The left side with respect to \( y \) and the right side with respect to \( x \):

\[
\int \frac{1}{y(y + 2)} \, dy = \int dx
\]

To integrate the left side, we'll use partial fraction decomposition.

### Step 3: Partial fraction decomposition

The expression \( \frac{1}{y(y + 2)} \) can be decomposed as:

\[
\frac{1}{y(y + 2)} = \frac{A}{y} + \frac{B}{y + 2}
\]

Multiplying both sides by \( y(y + 2) \):

\[
1 = A(y + 2) + By
\]

Expanding and combining like terms:

\[
1 = Ay + 2A + By
\]

\[
1 = (A + B)y + 2A
\]

This implies two equations:

\[
A + B = 0
\]
\[
2A = 1
\]

Solving these equations:

\[
A = \frac{1}{2}, \quad B = -\frac{1}{2}
\]

Thus, the partial fraction decomposition is:

\[
\frac{1}{y(y + 2)} = \frac{1/2}{y} - \frac{1/2}{y + 2}
\]

### Step 4: Integrate each term

Now we can integrate both sides:

\[
\int \left(\frac{1/2}{y} - \frac{1/2}{y + 2}\right) \, dy = \int dx
\]

Integrating:

\[
\frac{1}{2} \ln |y| - \frac{1}{2} \ln |y + 2| = x + C
\]

### Step 5: Simplify the result

Combine the logarithms:

\[
\frac{1}{2} \ln \left|\frac{y}{y + 2}\right| = x + C
\]

Exponentiating both sides to solve for \( y \):

\[
\ln \left|\frac{y}{y + 2}\right| = 2x + 2C
\]

\[
\left|\frac{y}{y + 2}\right| = e^{2x + 2C}
\]

Let \( e^{2C} = C_1 \), where \( C_1 \) is a new constant:

\[
\left|\frac{y}{y + 2}\right| = C_1 e^{2x}
\]

Removing the absolute value, we have two cases:

1. \(\frac{y}{y + 2} = C_1 e^{2x}\)
2. \(\frac{y}{y + 2} = -C_1 e^{2x}\)

These can be solved for \( y \), but the general solution in implicit form is:

\[
\frac{y}{y + 2} = C_1 e^{2x}
\]

If you'd like a more specific solution or further details, please let me know!

---

**Do you have any questions or would you like further details?**

**Here are 8 related questions that might interest you:**

1. How do you solve differential equations using other methods like the substitution method?
2. What are the real-world applications of solving separable differential equations?
3. Can we solve this equation using numerical methods? How would the solution differ?
4. What are the conditions for a differential equation to be considered separable?
5. How do initial conditions affect the solution of a differential equation?
6. What is the significance of the constant of integration in differential equations?
7. Can this differential equation be solved using a power series expansion?
8. How can Laplace transforms be used to solve differential equations?

**Tip:** When solving separable differential equations, always check if the function can be easily decomposed into partial fractions, as this simplifies the integration process.

Learn how to solve the separable differential equation y'(x) = y(x)(y(x) + 2) step-by-step. This problem covers key concepts in differential equations, including separable equations and partial fraction decomposition. Suitable for undergraduate-level study.

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