Verify by direct substitution that the given power series is a solution of the indicated differential equation. [Hint: For a power x2n + 1 let k = n + 1.] y = (-1) nx2n, (1 + x2)y' + 2xy = 0

The given power series is a solution of the differential equation (1+x^2)y' + 2xy = 0

Step-by-step explanation:

This is a very trivial exercise, follow the steps below for the solution:

Step 1: Since n = 0, 1, 2, 3, 4, ........, Substitute the values of n into equation (1) below.

.....................(1)

Step 2: Find the derivative of y, i.e. y'

Step 3: Substitute y and y' into equation (2) below:

(Verified)

Since the LHS = RHS = 0, the given power series is a solution of the differential equation (1+x^2)y' + 2xy = 0

450 is the final answer! : )