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

## 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′ + bx1 − 2ny = 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.

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

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