EqWorld The World of Mathematical Equations

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

## 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.

### 1.2. Other Linear Difference and Functional Equations

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

1. yn+2 + ayn+1 + byn = 0.    Second-order constant-coefficient linear homogeneous difference equation.
2. yn+2 + ayn+1 + byn = fn.    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

1. Ay(ax) + By(bx) + y(x) = 0.
2. Ay(xa) + By(xb) + y(x) = 0.
3. y(y(x)) − x = 0.     Babbage equation (equation of involutory functions).
4. y(y(x)) + ay(x) + bx = 0.
5. y(y(y(x))) − x = 0.
6. Ay(x) + By((ax − β)/(x + b)) + Cy((bx + β)/(a − x)) = f(x),    β = a2 + ab + b2.
7. f1(x)y(x) + f2(x)y((ax − β)/(x + b)) + f3(x)y((bx + β)/(a − x)) = g(x),    β = a2 + ab + b2.
8. yn+m + am−1yn+m−1 + ... + a1yn+1 + a0yn = 0.    mth-order constant-coefficient linear homogeneous difference equation.
9. yn+m + am−1yn+m−1 + ... + a1yn+1 + a0yn = fn.    mth-order constant-coefficient linear nonhomogeneous difference equation.
10. y(x + n) + an−1y(x + n − 1) + ... + a1y(x + 1) + a0y(x) = 0.    nth-order constant-coefficient linear homogeneous difference equation.
11. y(x + n) + an−1y(x + n − 1) + ... + a1y(x + 1) + a0y(x) = f(x).    nth-order constant-coefficient linear nonhomogeneous difference equation.
12. y(x + bn) + an−1y(x + bn−1) + ... + a1y(x + b1) + a0y(x) = 0.
13. ay(xα) + by(xβ) + cy(xσ) + ... + y(x) = 0.
14. y(anx) + bn−1y(an−1x) + ... + b1y(a1x) + b0y(x) = 0.
15. y[n](x) + an−1y[n−1](x) + ... + a1y(x) + a0x = 0,    y[n](x) = y(y[n−1](x)).

The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.