Integration Factor

The given differential equation is:

sin(x) * dx/dy + 2y * cos(x) = 1

To solve this equation, we can use an integrating factor. An integrating factor is a function that we multiply the entire equation by in order to make it an exact differential equation. In this case, we need to find a function u(x) such that when we multiply the equation by u(x), the left-hand side becomes the derivative of a product.

Deriving the Integration Factor

To find the integrating factor, we can rearrange the equation to isolate dx/dy:

dx/dy = (1 - 2y * cos(x)) / sin(x)

Now, let's compare this equation to the standard form of a linear differential equation:

dx/dy + P(x)y = Q(x)

In our case, P(x) = -2cos(x) and Q(x) = (1 - 2y * cos(x)) / sin(x).

The integrating factor, denoted by mu(x), is defined as:

mu(x) = e^(integral(P(x) dx))

Calculating the Integration Factor

In this case, P(x) = -2cos(x), so let's calculate the integral of -2cos(x) dx:

integral(-2cos(x) dx) = -2 * integral(cos(x) dx) = -2 * sin(x)

Therefore, the integrating factor mu(x) is:

mu(x) = e^(-2 * sin(x))

Using the Integration Factor

Now that we have the integrating factor, we multiply the entire equation by mu(x):

e^(-2 * sin(x)) * (sin(x) * dx/dy + 2y * cos(x)) = e^(-2 * sin(x))

Simplifying the left-hand side:

(sin(x) * e^(-2 * sin(x))) * dx/dy + 2y * cos(x) * e^(-2 * sin(x)) = e^(-2 * sin(x))

This can be written as the derivative of a product:

d/dy(sin(x) * e^(-2 * sin(x)) * y) = e^(-2 * sin(x))

Integrating and Solving for y

Integrating both sides with respect to y:

∫ d/dy(sin(x) * e^(-2 * sin(x)) * y) dy = ∫ e^(-2 * sin(x)) dy

sin(x) * e^(-2 * sin(x)) * y = ∫ e^(-2 * sin(x)) dy

y = ∫ (e^(-2 * sin(x)) / sin(x)) dy

Solving this integral would give us the solution for y, but in this case, we are only concerned with finding the integrating factor, which is sin^2(x) (option A).

Therefore, the correct answer is option A: sin^2(x).