SECTION 4.2 Constructing a Second Solution from a Known Solution 137 EXERCISES 4.2 Answers to odd-numbered problems begin on page A-7. In Problems 1-30 find a second solution of each differential equation. Use reduction of order or formula (4) as instructed. Assume an appropriate inter- val of validity. 1. y" + 5y' = 0; y1 = 1 2. y" - y' = 0; y1 = 1 3. y" - 4y' + 4y = 0; y1 = e2x 4. y" + 2y' + y = 0; y1 = xe"* 5. y" + 16y = 0; y1 = cos 4x 6. y" + 9y = 0; y1 = sin 3x 7. y" - y = 0; y1 = cosh x 8. y" - 25y = 0; y1 = esx 9. 9y" - 12y' + 4y = 0; y, = e2x/3 10. 6y" + y' - y = 0; y1 = ex/3 11. x2y" - 7xy' + 16y = 0; y1 = x4 12. x2y" + 2xy' - 6y = 0; y1 = x2 13. xy" + y' = 0; y1 = In x 14. 4x y" + y = 0; y, = x/2 In x 15. (1 - 2x - x2 )y" + 2(1 + x)y' - 2y = 0; y1 = x + 1 16. (1 - x2)y" - 2xy' = 0; y1 = 1 17. xy" - xy' + 2y = 0; y, = x sin(In x) ly" - 3xy' + 5y = 0; y1 = x2 cos (In x) 19. (1 + 2x)y" + 4xy' - 4y = 0; y = e-2x 20. (1 + x)y" + xy' - y = 0; y1 = x 21. xay" - xy' + y = 0; y1 = x 22. x2y" - 20y = 0; y1 = x 4 23. x2y" - 5xy' + 9y = 0; y, = x'Inx 24. x2y" + xy' + y = 0; y = cos(In x)