Obtain the general solution:
(2x^3 -2xy)dx = dy

y = ce^(-x^2) + x^2 - 1

y =ce^(x^2) + x^2 - 1

y =ce^(x^2) + x^2 + 1

y =ce^(-x^2) - x^2 - 1

The given differential equation is:

\[
(2x^3 - 2xy)dx = dy
\]

We will solve this step by step.

### Step 1: Rearranging the equation
Rewrite the equation in a more usable form by dividing both sides by \(dx\):

\[
2x^3 - 2xy = \frac{dy}{dx}
\]

### Step 2: Separation of variables
Group all terms involving \(y\) and \(dy\) on one side and those involving \(x\) on the other side. We have:

\[
\frac{dy}{dx} = 2x^3 - 2xy
\]

This is a first-order linear differential equation of the form:

\[
\frac{dy}{dx} + 2xy = 2x^3
\]

### Step 3: Solving using an integrating factor
The standard approach for solving this type of equation is to use an integrating factor \( \mu(x) \), which is given by:

\[
\mu(x) = e^{\int P(x) dx}
\]
where \( P(x) = 2x \). Thus, we compute the integrating factor:

\[
\mu(x) = e^{\int 2x dx} = e^{x^2}
\]

### Step 4: Multiply through by the integrating factor
Now, multiply both sides of the original equation by \( e^{x^2} \):

\[
e^{x^2} \frac{dy}{dx} + 2x e^{x^2} y = 2x^3 e^{x^2}
\]

The left-hand side is now the derivative of \( y e^{x^2} \):

\[
\frac{d}{dx} \left( y e^{x^2} \right) = 2x^3 e^{x^2}
\]

### Step 5: Integrate both sides
Integrate both sides with respect to \(x\):

\[
y e^{x^2} = \int 2x^3 e^{x^2} dx
\]

We can simplify the right-hand side by noticing that:

\[
\int 2x^3 e^{x^2} dx = e^{x^2}(x^2 - 1) + C
\]

### Step 6: Solve for \(y\)
Now, divide both sides by \( e^{x^2} \):

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

Thus, the general solution is:

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

### Conclusion:
The correct answer is:

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

This corresponds to option 1:
\[
y = ce^{-x^2} + x^2 - 1
\]

---

Would you like further clarification on any of the steps or more details?

Here are 5 related questions to explore further:

1. How do you find integrating factors for different types of differential equations?
2. What is the general method for solving first-order linear differential equations?
3. Can every first-order differential equation be solved by separation of variables?
4. What are the conditions for the existence of an integrating factor?
5. How do exponential functions affect the behavior of solutions in differential equations?

**Tip**: Always check if the differential equation can be simplified or factored before attempting more complex methods like integrating factors.

Obtain the general solution: (2x^3 - 2xy)dx = dy

This guide provides a detailed solution to the first-order linear differential equation (2x^3 - 2xy)dx = dy. Learn step-by-step how to use separation of variables and integrating factors to obtain the general solution: y = C e^{-x^2} + x^2 - 1. Suitable for undergraduate students studying calculus or differential equations.

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