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

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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.

2.2. Ordinary Differential Equations Involving Exponential and Other Functions

  1. y″ + aeλxy = 0.
  2. y″ + (aex − b)y = 0.
  3. y″ − (aex + beλx + c)y = 0.
  4. y″ + ay′ + be2axy = 0.
  5. y″ − ay′ + be2axy = 0.
  6. y″ + ay′ + (beλx + c)y = 0.
  7. y″ − (a − 2q cosh 2x)y = 0.    Modified Mathieu equation.
  8. y″ + (a − 2q cos 2x)y = 0.    Mathieu equation.
  9. y″ + a tan x y′ + by = 0.

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

  1. y″ + fy′ + a(f − a)y = 0.
  2. y″ + xfy′ − fy = 0.
  3. xy″ + (xf + a)y′ + (a − 1)fy = 0.
  4. xy″ + [(ax + 1)f + ax − 1]y′ + a2xfy = 0.
  5. xy″ + [(ax2 + bx)f + 2]y′ + bfy = 0.
  6. x2y″ + xfy′ + a(f − a − 1)y = 0.
  7. y″ + (f + aeλx)y′ + aeλx(f + λ)y = 0.
  8. y″ − (f 2 + f′)y = 0.
  9. y″ + 2fy′ + (f 2 + f′)y = 0.
  10. y″ + (1 − a)fy′ − a(f 2 + f′)y = 0.
  11. y″ + fy′ + (fg − g2 + g′)y = 0,   g=g(x).
  12. fy″ − af′y′ − bf 2a + 1y = 0.
  13. f 2y″ + f(f′ + a)y′ + by = 0.
  14. y″ − f′y′ + a2e2fy = 0.
  15. y″ − f′y′ − a2e2fy = 0.

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