Exact Solutions > Functional Equations > Linear Difference and Functional Equations with One Independent Variable

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1. Linear Difference and Functional Equations with One Independent Variable

1.1. Linear Difference and Functional Equations Containing Unknown Function with Two Different Arguments

First-order linear difference equations

1. y(x + 1) − ay(x) = 0.    First-order constant-coefficient linear homogeneous difference equation.
2. y(x + 1) − ay(x) = f(x).    First-order constant-coefficient linear nonhomogeneous difference equation.
3. y(x + 1) − xy(x) = 0.
4. y(x + 1) − a(x − b)(x − c)y(x) = 0.
5. y(x + 1) − R(x)y(x) = 0,   where  R(x) is a rational function.
6. y(x + 1) − f(x)y(x) = 0.
7. y(x + a) − by(x) = 0.
8. y(x + a) − by(x) = f(x).
9. y(x + a) − bxy(x) = 0.
10. y(x + a) − f(x)y(x) = 0.

Linear functional equations containing y(x) and y(ax)

11. y(ax) − by(x) = 0.
12. y(ax) − by(x) = f(x).

Linear functional equations containing y(x) and y(a − x)

13. y(x) − y(a − x) = 0.
14. y(x) + y(a − x) = 0.
15. y(x) + y(a − x) = b.
16. y(x) + y(a − x) = f(x).
17. y(x) − y(a − x) = f(x).
18. y(x) + g(x)y(a − x) = f(x).

Linear functional equations containing y(x) and y(z), where z = φ(x)

19. y(x^a) − by(x) = 0.
20. y(x) − y(a/x) = 0.
21. y(x) + y(a/x) = 0.
22. y(x) + y(a/x) = b.
23. y(x) + y(a/x) = f(x).
24. y(x) − y(a/x) = f(x).
25. y(x) + g(x)y(a/x) = f(x).
26. y(x) − y((a − x)/(1 + bx)) = 0.
27. y(x) + y((a − x)/(1 + bx)) = 0.
28. y(x) + y((a − x)/(1 + bx)) = f(x).
29. y(x) − y((a − x)/(1 + bx)) = f(x).
30. y(x) − cy((a − x)/(1 + bx)) = f(x).
31. y(x) + g(x)y((a − x)/(1 + bx)) = f(x).
32. y(x) + cy((ax − β)/(x + b)) = f(x),    β = a² + ab + b².
33. y(x) + cy((bx + β)/(a − x)) = f(x),    β = a² + ab + b².
34. y(x) + g(x)y((ax − β)/(x + b)) = f(x),    β = a² + ab + b².
35. y(x) + g(x)y((bx + β)/(a − x)) = f(x),    β = a² + ab + b².
36. y(x) − y((a² − x²)^1/2) = 0.
37. y(x) + y((a² − x²)^1/2) = 0.
38. y(x) + y((a² − x²)^1/2) = b.
39. y(x) + y((a² − x²)^1/2) = f(x).
40. y(x) − y((a² − x²)^1/2) = f(x).
41. y(x) + g(x)y((a² − x²)^1/2) = f(x).

Linear functional equations containing y(sin x) and y(cos x)

42. y(sin x) − y(cos x) = 0.
43. y(sin x) + y(cos x) = 0.
44. y(sin x) + y(cos x) = a.
45. y(sin x) + y(cos x) = f(x).
46. y(sin x) − y(cos x) = f(x).
47. y(sin x) + g(x)y(cos x) = f(x).

Linear functional equations containing y(x) and y(ω(x)), where ω(ω(x)) = x

48. y(x) − y(ω(x)) = 0,    where   ω(ω(x)) = x.
49. y(x) + y(ω(x)) = 0,    where   ω(ω(x)) = x.
50. y(x) + y(ω(x)) = b,    where   ω(ω(x)) = x.
51. y(x) + y(ω(x)) = f(x),    where   ω(ω(x)) = x.
52. y(x) − y(ω(x)) = f(x),    where   ω(ω(x)) = x.
53. y(x) + g(x)y(ω(x)) = f(x),    where   ω(ω(x)) = x.

1.2. Other Linear Difference and Functional Equations

Second-order linear difference equations, y_n = y(n)

1. y_n+2 + ay_n+1 + by_n = 0.    Second-order constant-coefficient linear homogeneous difference equation.
2. y_n+2 + ay_n+1 + by_n = f_n.    Second-order constant-coefficient linear nonhomogeneous difference equation.
3. y(x + 2) + ay(x + 1) + by(x) = 0.    Second-order constant-coefficient linear homogeneous difference equation.
4. y(x + 2) + ay(x + 1) + by(x) = f(x).    Second-order constant-coefficient linear nonhomogeneous difference equation.
5. y(x + 2) + a(x + 1)y(x + 1) + bx(x + 1)y(x) = 0.

Other functional equations

6. Ay(ax) + By(bx) + y(x) = 0.
7. Ay(x^a) + By(x^b) + y(x) = 0.
8. y(y(x)) − x = 0.     Babbage equation (equation of involutory functions).
9. y(y(x)) + ay(x) + bx = 0.
10. y(y(y(x))) − x = 0.
11. Ay(x) + By((ax − β)/(x + b)) + Cy((bx + β)/(a − x)) = f(x),    β = a² + ab + b².
12. f₁(x)y(x) + f₂(x)y((ax − β)/(x + b)) + f₃(x)y((bx + β)/(a − x)) = g(x),    β = a² + ab + b².
13. y_n+m + a_m−1y_n+m−1 + ... + a₁y_n+1 + a₀y_n = 0.    mth-order constant-coefficient linear homogeneous difference equation.
14. y_n+m + a_m−1y_n+m−1 + ... + a₁y_n+1 + a₀y_n = f_n.    mth-order constant-coefficient linear nonhomogeneous difference equation.
15. y(x + n) + a_n−1y(x + n − 1) + ... + a₁y(x + 1) + a₀y(x) = 0.    nth-order constant-coefficient linear homogeneous difference equation.
16. y(x + n) + a_n−1y(x + n − 1) + ... + a₁y(x + 1) + a₀y(x) = f(x).    nth-order constant-coefficient linear nonhomogeneous difference equation.
17. y(x + b_n) + a_n−1y(x + b_n−1) + ... + a₁y(x + b₁) + a₀y(x) = 0.
18. ay(x^α) + by(x^β) + cy(x^σ) + ... + y(x) = 0.
19. y(a_nx) + b_n−1y(a_n−1x) + ... + b₁y(a₁x) + b₀y(x) = 0.
20. y^[n](x) + a_n−1y^[n−1](x) + ... + a₁y(x) + a₀x = 0,    y^[n](x) = y(y^[n−1](x)).