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?

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.