Final answer:
The differential equation (x) dy/dx + (sin(x))y = 1 can be solved by finding an integrating factor e∫ sin(x)/x dx and applying the method of integrating factors to obtain the general solution.
Step-by-step explanation:
To find the general solution of the differential equation (x) dy/dx + (sin(x))y = 1, we should use the method of integrating factors. This method involves multiplying the entire equation by a function that will allow us to write the left side as the derivative of a product.
Firstly, let's find the integrating factor, μ(x), which is given by e∫ P(x) dx, where P(x) is the coefficient of y. In our case, P(x) = sin(x)/x, thus the integrating factor μ(x) = e∫ sin(x)/x dx.
Note, that the integral of sin(x)/x dx cannot be expressed in terms of elementary functions, so the integrating factor itself cannot be simplified further. However for the purpose of solving the differential equation, we don't necessarily need the explicit form of the integrating factor.
After multiplying the original equation by μ(x), we should be able to write the left side as (d/dx)[μ(x)y(x)] and our equation will take the form (d/dx)[μ(x)y(x)] = μ(x). Integrating both sides with respect to x gives us μ(x)y(x) = ∫ μ(x) dx + C, where C is the constant of integration.
Finally, the general solution y(x) can be expressed implicitly as y(x) = (∫ μ(x) dx + C) / μ(x).