Exact Solutions > Ordinary Differential Equations > Second-Order Linear Ordinary Differential Equations

PDF version of this page

2. Second-Order Linear Ordinary Differential Equations

2.1. Ordinary Differential Equations Involving Power Functions

1. y″ + ay = 0.    Equation of free oscillations.
2. y″ − axⁿy = 0.
3. y″ + ay′ + by = 0.    Second-order constant coefficient linear equation.
4. y″ + ay′ + (bx + c)y = 0.
5. y″ + (ax+ b)y′ + (αx² + βx + σ)y = 0.
6. xy″ + ay′ + by = 0.
7. xy″ + ay′ + bxy = 0.
8. xy″ + ny′ + bx^{1 − 2n}y = 0.
9. xy″ + ay′ + bxⁿy = 0.
10. xy″ + (b − x)y′ − ay = 0.    Degenerate hypergeometric equation.
11. (a₂x + b₂)y″ + (a₁x + b₁)y′ + (a₀x + b₀)y = 0.
12. x²y″ + axy′ + by = 0.    Euler equation.
13. x²y″ + xy′ + (x² − ν²)y = 0.    Bessel equation.
14. x²y″ + xy′ − (x² + ν²)y = 0.    Modified Bessel equation.
15. x²y″ + axy′ + (bxⁿ + c)y = 0.
16. x²y″ + axy′ + xⁿ(bxⁿ + c)y = 0.
17. x²y″ + (ax + b)y′ + cy = 0.
18. (1 − x²)y″ − 2xy′ + n(n + 1)y = 0,   n = 0, 1, 2, ...    Legendre equation.
19. (1 − x²)y″ − 2xy′ + ν(ν + 1)y = 0.    Legendre equation.
20. (ax² + b)y″ + axy′ + cy = 0.
21. (1 − x²)y″ + (ax + b)y′ + cy = 0.
22. x(x − 1)y″ + [(α + β + 1)x − γ]xy′ + αβy = 0.    Gaussian hypergeometric equation.
23. (1 − x²)²y″ − 2x(1 − x²)y′ + [ν(ν + 1)(1 − x²) − μ²]y = 0.    Legendre equation.
24. (x − a)²(x − b)²y″ − cy = 0.
25. (ax² + bx + c)²y″ + Ay = 0.
26. x²(axⁿ − 1)y″ + x(apxⁿ + q)y′ + (arxⁿ + s)y = 0.

2.2. Ordinary Differential Equations Involving Exponential and Other Functions

27. y″ + ae^λxy = 0.
28. y″ + (ae^x − b)y = 0.
29. y″ − (ae^2λx + be^λx + c)y = 0.
30. y″ + ay′ + be^2axy = 0.
31. y″ − ay′ + be^2axy = 0.
32. y″ + ay′ + (be^λx + c)y = 0.
33. y″ − (a − 2q cosh 2x)y = 0.    Modified Mathieu equation.
34. y″ + (a − 2q cos 2x)y = 0.    Mathieu equation.
35. y″ + a tan x y′ + by = 0.

2.3. Ordinary Differential Equations Involving Arbitrary Functions, f = f(x)

36. y″ + fy′ + a(f − a)y = 0.
37. y″ + xfy′ − fy = 0.
38. xy″ + (xf + a)y′ + (a − 1)fy = 0.
39. xy″ + [(ax + 1)f + ax − 1]y′ + a²xfy = 0.
40. xy″ + [(ax² + bx)f + 2]y′ + bfy = 0.
41. x²y″ + xfy′ + a(f − a − 1)y = 0.
42. y″ + (f + ae^λx)y′ + ae^λx(f + λ)y = 0.
43. y″ − (f ² + f′)y = 0.
44. y″ + 2fy′ + (f ² + f′)y = 0.
45. y″ + (1 − a)fy′ − a(f ² + f′)y = 0.
46. y″ + fy′ + (fg − g² + g′)y = 0,   g=g(x).
47. fy″ − af′y′ − bf ^{2a + 1}y = 0.
48. f ²y″ + f(f′ + a)y′ + by = 0.
49. y″ − f′y′ + a²e^2fy = 0.
50. y″ − f′y′ − a²e^2fy = 0.