Reduce the differential equation below to an exact equation

(X - 2siny + 3)dx -(4siny - 2x + 3)dy=0. The problem here is how to solve for the integrating factor...any help pls?

that siny in both factors is bad news. I'd expect one of them to have cosy. Sure there's no typo?

Also the lone x is not good. There ought to be an x^2 in the dy factor.

Also, the form is usually

M dx + N dy = 0
not M dx - N dy = 0

This whole one is weird.

To determine if the given differential equation can be reduced to an exact equation, we need to check if the equation satisfies the following condition:

Mdx + Ndy = 0, where M = X - 2siny + 3 and N = -(4siny - 2x + 3)

To check for exactness, we need to verify if ∂M/∂y = ∂N/∂x. Let's calculate the partial derivatives:

∂M/∂y = -2cosy
∂N/∂x = -2

Since ∂M/∂y ≠ ∂N/∂x, the equation is not exact.

To make the equation exact, we need to find an integrating factor. The integrating factor can be calculated using the formula:

μ = e^(∫(∂M/∂y - ∂N/∂x)dx)

In this case, (∂M/∂y - ∂N/∂x) = -2cosy - (-2) = -2cosy + 2

∫(-2cosy + 2)dx = -2xsin(y) + 2x

Therefore, the integrating factor μ can be obtained as:

μ = e^(-2xsin(y) + 2x)

Multiplying the given equation by the integrating factor μ, we get:

μ * (X - 2siny + 3)dx - μ * (4siny - 2x + 3)dy = 0

Simplifying this equation will result in an exact differential equation.

To reduce the given differential equation to an exact equation, we need to identify an integrating factor. Here's how to solve for it:

Step 1: Check for Exactness
First, check if the equation is already exact by ensuring the mixed partial derivatives are equal. In this case, calculate the partial derivative of (X - 2siny + 3) with respect to x and the partial derivative of (4siny - 2x + 3) with respect to y. If these partial derivatives are equal (∂(X - 2siny + 3)/∂x = ∂(4siny - 2x + 3)/∂y), then the equation is already exact and no further steps are needed.

Step 2: Find the Integrating Factor
If the equation is not exact, we need to find the integrating factor. The integrating factor can be determined by dividing the coefficient of dy by the coefficient of dx and then integrating that expression. In this case, the coefficient of dy is (4siny - 2x + 3), and the coefficient of dx is (X - 2siny + 3).

Divide the coefficients: (4siny - 2x + 3) / (X - 2siny + 3)

Step 3: Simplify the Integrating Factor
To simplify the integrating factor, we need to express it as a function of x or y only. In this case, let's express it as a function of x.

Simplify the integrating factor expression by multiplying the numerator and denominator by -1: (2x - 4siny - 3) / (2siny - X + 3)

Step 4: Check for Exactness Again
Multiply the entire given equation by the integrating factor obtained in Step 3. The result should be an exact equation. If not, there might have been a sign error or simplification issue in the integrating factor calculation.

Step 5: Solve the Exact Equation
Once the equation is exact after multiplying by the integrating factor, solve it by finding the antiderivative of the coefficients with respect to x and y. The solution will include a constant term.

By following these steps, you should be able to reduce the given differential equation to an exact equation.