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The problem provided is a differential equation along with an initial condition:

\[
y' = -2xy^2, \quad y(0) = 5
\]

We are tasked with solving this differential equation and finding the particular solution that satisfies the given initial condition.

### Step 1: Separate the Variables

To solve this differential equation, we will use the method of separation of variables.

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

We can rewrite this as:

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

### Step 2: Integrate Both Sides

Integrating both sides:

\[
\int \frac{dy}{y^2} = \int -2x \, dx
\]

The left side integrates to:

\[
\int y^{-2} \, dy = -\frac{1}{y} + C_1
\]

The right side integrates to:

\[
\int -2x \, dx = -x^2 + C_2
\]

Thus, the equation becomes:

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

where \( C = C_2 - C_1 \) is a constant.

### Step 3: Solve for \( y \)

Solving for \( y \):

\[
\frac{1}{y} = x^2 - C
\]

\[
y = \frac{1}{C - x^2}
\]

### Step 4: Apply the Initial Condition

We use the initial condition \( y(0) = 5 \) to find the constant \( C \).

\[
5 = \frac{1}{C - 0^2}
\]

\[
5 = \frac{1}{C}
\]

\[
C = \frac{1}{5}
\]

### Step 5: Write the Particular Solution

The particular solution to the differential equation is:

\[
y = \frac{1}{\frac{1}{5} - x^2} = \frac{5}{1 - 5x^2}
\]

This is the solution that satisfies the initial condition \( y(0) = 5 \).

### Conclusion
The correct answer is the expression \( y = \frac{5}{1 - 5x^2} \).

Would you like more details on any of the steps? Here are five related questions:

1. How do you solve differential equations using the method of separation of variables?
2. What other methods can be used to solve first-order differential equations?
3. Can you explain why integrating both sides is necessary in solving differential equations?
4. What would happen if the initial condition was different, say \( y(0) = 3 \)?
5. How does the constant of integration affect the general solution of a differential equation?

**Tip:** When dealing with differential equations, always check if the problem allows you to separate variables. If not, consider alternative methods like integrating factors or substitution.

Ache a solução da equação diferencial y' = -2xy^2, que atenda à condição inicial y(0) = 5 e assinale a alternativa que a contenha:

Learn how to solve the differential equation y' = -2xy^2 with the initial condition y(0) = 5 using separation of variables. Detailed step-by-step solution including integration techniques and application of the initial condition. Suitable for undergraduate math students.

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