Linear Difference Equations

Mathematical Modeling · Axiom Academy

LESSON Linear Difference Equations Master the techniques for solving linear difference equations with constant coefficients, from first-order recurrences to the characteristic equation method for higher orders. 1. First-Order Homogeneous Equations The simplest linear difference equation is the first-order homogeneous equation: This equation says that each term is a constant multiple of the previous term. The solution follows directly from repeated application of the recurrence. 2. First-Order Non-Homogeneous Equations Adding a constant term gives us the non-homogeneous first-order equation: The solution combines two parts: the homogeneous solution (what happens without b) and a particular solution (a fixed point of the system). 3. Second-Order Linear Equations Second-order linear difference equations involve two previous terms: The key insight is to guess a solution of the form x_n = r^n for some constant r. Substituting this guess leads to the characteristic equation . The discriminant of the characteristic equation determines which case applies: When r_1 != r_2, the general solution is: When r_1 = r_2 = r, we need an extra factor of n: When roots are complex conjugates, use polar form:

This is the written version of the interactive lesson above. See the full Mathematical Modeling course.