Integrating Factor of the Equation

To find the integrating factor of the given equation, we will use the method of integrating factors. The integrating factor is a function that helps in solving linear first-order differential equations. In this case, we have the equation:

dy/dx + x sin(2y) = x^3 cos^2(y)

To find the integrating factor, we need to multiply the entire equation by a suitable function f(x). Let's assume the integrating factor is f(x). Multiplying the equation by f(x), we get:

f(x) dy/dx + f(x) x sin(2y) = f(x) x^3 cos^2(y)

Now we need to choose f(x) such that the left-hand side of the equation becomes an exact differential. An exact differential is a differential form that can be expressed as the derivative of a function. In this case, we want to make the left-hand side of the equation of the form d(ψ)/dx, where ψ is some function of x and y.

Using the Integrating Factor

To find the integrating factor, we can compare the terms involving dy. The term f(x) dy/dx should match the form d(ψ)/dx. In this case, the term f(x) x sin(2y) should be the derivative of some function with respect to x.

Comparing the terms, we can see that f(x) x sin(2y) should be the derivative of ψ with respect to x. In other words:

d(ψ)/dx = f(x) x sin(2y)

Comparing this equation with the chain rule in calculus, we can deduce that:

∂ψ/∂x = f(x) x sin(2y)

To simplify the equation, we can assume that ψ only depends on y. Therefore, the partial derivative of ψ with respect to x is zero (∂ψ/∂x = 0). This gives us:

0 = f(x) x sin(2y)

Finding the Value of f(0)

To find the value of f(0), we substitute x = 0 into the equation. This gives us:

0 = f(0) * 0 * sin(2y)

Since sin(2y) can take any value, we can conclude that f(0) can be any constant value. However, we are asked to find the correct value of f(0), which is 1.

Therefore, the correct answer is f(0) = 1.