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TWK2A Reduction of order (Section 4.2) Problems 1. If y1 = e5x is a solution of y 00 second solution. 25y = 0; use reduction of order to …nd a 2. If y1 = x4 is a solution of x2 y 00 …nd a second solution. 7xy 0 + 16y = 0; use reduction of order to 3. If y1 = ln x is a solution of xy 00 + y 0 = 0; use reduction of order to …nd a second solution. 4. If y1 = x2 cos (ln x) is a solution of x2 y 00 order to …nd a second solution. x 5. If y1 = e 3 is a solution of 6y 00 + y 0 second solution. 3xy 0 + 5y = 0; use reduction of y = 0; use reduction of order to …nd a 6. If y1 (x) is a solution to a linear homogeneous second-order DE, a second solution to the equation is given by R Z e P (x)dx dx; y2 (x) = y1 (x) y12 (x) where P (x) is the coe¢ cient of y 0 in the standard form of the ODE. Show that y1 and y2 are linearly independent. 7. Consider the equation ay 00 + by 0 + cy = 0: b Given that y1 (x) = e 2a x is a solution to this DE, use reduction of order to …nd a second solution.