The differential equation given is:

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

This is a separable differential equation, meaning we can separate the variables \(x\) and \(y\) on different sides of the equation. Let's solve it step by step.

### Step 1: Separate the variables
First, we'll rearrange the equation to separate \(y\) and \(x\):

\[
y^2 \, dy = 2x \, dx
\]

### Step 2: Integrate both sides
Next, we'll integrate both sides of the equation:

\[
\int y^2 \, dy = \int 2x \, dx
\]

### Step 3: Perform the integration
Let's integrate both sides:

- The left side:

\[
\int y^2 \, dy = \frac{y^3}{3} + C_1
\]

- The right side:

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

### Step 4: Combine the constants
Since both sides have an arbitrary constant, we can combine them into a single constant \(C\):

\[
\frac{y^3}{3} = x^2 + C
\]

### Step 5: Solve for \( y \)
To express \(y\) explicitly in terms of \(x\), multiply both sides by 3:

\[
y^3 = 3x^2 + 3C
\]

Taking the cube root:

\[
y = \sqrt[3]{3x^2 + 3C}
\]

Here, \(C\) is an arbitrary constant that can be redefined as a new constant \(C' = 3C\), giving us:

\[
y = \sqrt[3]{3x^2 + C'}
\]

This is the general solution to the differential equation.

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

Here are 5 related questions to consider:
1. How would the solution change if the differential equation was \( \frac{dy}{dx} = 3xy^{-2} \)?
2. Can you solve \( \frac{dy}{dx} = y^2 \) and compare it with this problem?
3. What if the differential equation was \( \frac{dy}{dx} = 2xy^{-3} \)?
4. How does the presence of \( y^{-2} \) affect the nature of the solution?
5. Can you find the particular solution if an initial condition, such as \( y(0) = 1 \), is provided?

**Tip:** When dealing with separable differential equations, always check if the variables can be separated easily, as it simplifies the integration process.

Learn how to solve the differential equation dy/dx = 2xy^-2 with a detailed step-by-step solution. This problem covers key concepts in differential equations, including separable differential equations and integration techniques. Suitable for advanced undergraduate students.

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