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Follow the steps below to use the method of reduction of order to find a second...

Follow the steps below to use the method of reduction of order to find a second solution y2 given the following differential equation and y1, which solves the given homogeneous equation:

xy" + y' = 0; y1 = ln(x)

Step #1: Let y2 = uy1, for u = u(x), and find y'2 and y"2.

Step #2: Plug y'2 and y"2 into the differential equation and simplify.

Step #3: Use w = u' to transform your previous answer into a linear first order differential equation in w.

Step #4: Solve for w and thus u' in your previous answer.

Step #5: Solve for u from your previous equation for u'.

Step #6: Solve for y2 using our definition in the first step, (use c1 = -1 and c2 = 0, which are assumed to be the constants of integration).

Step #7: Verify that y2 is a solution to the original given differential equation, and check that W[y1, y2] does not equal zero in I := R.

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