The given differential equation is:

$y' = (2y - 1) \sin(x) \cos(x)$

This is a first-order linear differential equation where the right-hand side involves both $y$ and $x$. To solve this equation, we can attempt separation of variables. Let's go through the steps:

Step 1: Simplify the right-hand side

Recall the trigonometric identity: $\sin(2x) = 2 \sin(x) \cos(x)$ Thus, we can rewrite the equation as: $y' = (2y - 1) \frac{1}{2} \sin(2x)$ or $y' = \frac{1}{2} (2y - 1) \sin(2x)$

Step 2: Separation of variables

We want to isolate the terms involving $y$ on one side and those involving $x$ on the other side.

$\frac{dy}{dx} = \frac{1}{2} (2y - 1) \sin(2x)$

Now, divide both sides by $2y - 1$ and multiply both sides by $dx$:

$\frac{dy}{2y - 1} = \frac{1}{2} \sin(2x) dx$

Step 3: Integrate both sides

Next, we integrate both sides:

1. For the left-hand side: The integral of $\frac{1}{2y - 1}$ is: $\int \frac{dy}{2y - 1} = \frac{1}{2} \ln |2y - 1|$

2. For the right-hand side: The integral of $\frac{1}{2} \sin(2x)$ is: $\int \frac{1}{2} \sin(2x) \, dx = -\frac{1}{4} \cos(2x)$

So, after integrating both sides, we get:

$\frac{1}{2} \ln |2y - 1| = -\frac{1}{4} \cos(2x) + C$

Step 4: Solve for $y$

Multiply both sides of the equation by 2:

$\ln |2y - 1| = -\frac{1}{2} \cos(2x) + 2C$

Now, exponentiate both sides to remove the logarithm:

$|2y - 1| = e^{-\frac{1}{2} \cos(2x) + 2C} = e^{2C} e^{-\frac{1}{2} \cos(2x)}$

Let $e^{2C} = A$ (where $A$ is a constant), so the equation becomes:

$|2y - 1| = A e^{-\frac{1}{2} \cos(2x)}$

Finally, remove the absolute value by introducing a constant $B$ (which can be positive or negative):

$2y - 1 = B e^{-\frac{1}{2} \cos(2x)}$

Solve for $y$:

$y = \frac{1}{2} + \frac{B}{2} e^{-\frac{1}{2} \cos(2x)}$

Final Solution

The general solution to the differential equation is:

$y = \frac{1}{2} + \frac{B}{2} e^{-\frac{1}{2} \cos(2x)}$

Where $B$ is an arbitrary constant.

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Would you like more details on any part of the solution, or do you have any questions about this process?

Here are 5 related questions to help deepen your understanding:

1. How can you determine the value of the constant $B$ if an initial condition is provided?
2. What are other methods for solving first-order differential equations?
3. What happens if you have a non-linear differential equation like $y' = y^2 \sin(x)$?
4. How can you check if the solution $y = \frac{1}{2} + \frac{B}{2} e^{-\frac{1}{2} \cos(2x)}$ is correct by substitution?
5. How do solutions to differential equations relate to the behavior of physical systems like population models or radioactive decay?

Tip: Always simplify the right-hand side of the differential equation as much as possible before attempting separation of variables. It can make integration easier!