Question

Using "Integrating Factor" Method Find The General Solution To: 1/Xdy/1dx−2y/x²=Xcosx 2- For Below Initial Value Problem

Answer 1

The general solution to the differential equation is obtained by using the Integrating Factor method, which involves finding an integrating factor, making the left side of the differential equation a perfect derivative, and then integrating to solve for y.

Step-by-step explanation:

To find the general solution to the differential equation 1/x dy/dx − 2y/x² = x cos(x) using the Integrating Factor method, we proceed as follows:

1. Recognize that it is a first-order linear non-homogeneous differential equation in the form dy/dx + P(x)y = Q(x).
2. The integrating factor, μ(x), is given by e∫ P(x)dx. For the equation 1/x dy/dx − 2y/x² = x cos(x), P(x) = − 2/x. Hence, μ(x) = e∛ (-2/x)dx = eln(x−2) = x−2.
3. Multiply both sides of the differential equation by μ(x) to make the left side a perfect derivative of the product μ(x)y.
4. Integrate the resulting equation with respect to x to find y, which will include an arbitrary constant C.
5. Apply the initial conditions if given to solve for C and obtain the particular solution.

By applying these steps, you'll be able to work through the Integrating Factor method to solve the given differential equation and find the general or particular solutions.