differential equation

In Problems 1 –16 the indicated function y1(x) is a solution
of the given differential equation. Use reduction of order or
formula (5), as instructed, to find a second solution y2(x).
1. y” – 4y’ + 4y = 0; y1 = e2x
2. y” + 2y’ + y = 0; y1 = xe–x
3. y” + 16y = 0; y1 = cos 4x
4. y” + 9y = 0; y1 = sin 3x
5. y” – y = 0; y1 = cosh x
6. y” – 25y = 0; y1 = e5x
7. 9y” – 12y’ + 4y = 0; y1 = e2x/3
8. 6y” + y’ – y = 0; y1 = ex/3
9. x2y” – 7xy’ + 16y = 0; y1 = x4
10. x2y” + 2xy’ – 6y = 0; y1 = x2
11. xy” + y’ = 0; y1 = ln x
12. 4x2y” + y = 0; y1 = x1/2 ln x
13. x2y” – xy’ + 2y = 0; y1 = x sin(ln x)
14. x2y” – 3xy’ + 5y = 0; y1 = x2 cos(ln x)
15. (1 – 2x – x2)y” + 2(1 + x)y’ – 2y = 0; y1 = x + 1

16. (1 – x2)y” + 2xy’ = 0; y1 = 1

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