https://file.aiquickdraw.com/mathbot/file/17273427787746gjkrsa5.jpg

The equation shown in the image is a first-order linear differential equation:

\[
\frac{dy}{dx} = xy - y
\]

### Step-by-Step Solution:

1. **Simplify the equation**: We can factor the right-hand side to make the equation easier to solve.

\[
\frac{dy}{dx} = y(x - 1)
\]

2. **Separate the variables**: The equation is separable, meaning we can separate the terms involving \(y\) on one side and those involving \(x\) on the other side.

\[
\frac{1}{y} \, dy = (x - 1) \, dx
\]

3. **Integrate both sides**: Integrating both sides will help us find the general solution.

\[
\int \frac{1}{y} \, dy = \int (x - 1) \, dx
\]

The integral of \(\frac{1}{y}\) is \(\ln |y|\), and the integral of \(x - 1\) is:

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

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

4. **Solve for \(y\)**: Exponentiate both sides to eliminate the natural logarithm and solve for \(y\).

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

This simplifies to:

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

where \(C\) is now an arbitrary constant that can absorb the absolute value.

### Final Solution:
The general solution to the differential equation is:

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

### Summary:
We separated the variables, integrated both sides, and solved for \(y\) to find the general solution to the differential equation. The solution involves an arbitrary constant \(C\), representing the family of solutions to this first-order differential equation.

Would you like any further details or explanations on specific steps?

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Here are five related questions you might consider:
1. How do we handle initial conditions for this solution?
2. What is the significance of the constant \(C\) in the solution?
3. Can this method be applied to non-linear differential equations?
4. What happens if we are given a boundary condition at a specific point \(x\)?
5. How does separation of variables differ from other methods of solving differential equations?

**Tip**: Always verify that a solution satisfies the original differential equation by plugging it back into the equation.

Learn how to solve the first-order differential equation dy/dx = xy - y using separation of variables. This step-by-step solution involves integrating both sides and exponentiating to find the general solution y = Ce^(x^2/2 - x). Suitable for undergraduate students studying differential equations.

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