The given equation is:

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

We need to solve for \( x \) in terms of \( y \).

### Step 1: Rearrange the equation
Start by isolating the term involving \( x \) on one side:

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

### Step 2: Solve the differential equation
This is a first-order linear differential equation. To solve it, we can use an integrating factor. The equation has the form:

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

Where the coefficient of \( x \) is \( -\frac{1}{y} \).

### Step 3: Find the integrating factor
The integrating factor \( \mu(y) \) is given by:

\[
\mu(y) = e^{\int -\frac{1}{y} \, dy} = e^{-\ln|y|} = \frac{1}{y}
\]

### Step 4: Multiply through by the integrating factor
Multiply both sides of the differential equation by \( \frac{1}{y} \):

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

This simplifies to:

\[
\frac{d}{dy} \left( \frac{x}{y} \right) = 2
\]

### Step 5: Integrate both sides
Now integrate both sides with respect to \( y \):

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

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

### Step 6: Solve for \( x \)
Multiply through by \( y \) to solve for \( x \):

\[
x = 2y^2 + Cy
\]

Thus, the general solution is:

\[
x(y) = 2y^2 + Cy
\]

Would you like more details, or do you have any specific questions?

### Related Questions:
1. What is the method of integrating factors for solving first-order linear differential equations?
2. Can we solve this equation if initial conditions for \( x \) or \( y \) are provided?
3. How do you verify the solution of a differential equation?
4. What are the different types of differential equations?
5. How can we interpret the solution \( x = 2y^2 + Cy \) geometrically?

### Tip:
When solving differential equations, always check if the equation can be simplified or if an integrating factor can be applied to make the solution process easier.

Learn how to solve the first-order linear differential equation y(dx/dy) - x = 2y^2 using integrating factors. This solution covers key concepts such as first-order linear equations, the method of integrating factors, and provides a step-by-step guide to solving the equation. Suitable for undergraduate calculus students.

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