The Characteristic Equation

Differential Equations · Axiom Academy

LESSON The Characteristic Equation Assume a solution of the form y = e^ rx and a calculus problem turns into an algebra problem. 1. The Big Idea: Exponential Substitution Consider a second-order linear homogeneous differential equation with constant coefficients: The key insight: assume a solution of the form y = e^ rx , where r is an unknown constant we need to find. 2. Deriving the Characteristic Equation Substitute y = e^ rx , y' = re^ rx , and y'' = r^2e^ rx into ay'' + by' + cy = 0 and watch the calculus cancel out: 3. Three Cases: The Nature of the Roots The characteristic equation ar^2 + br + c = 0 is just a quadratic in r . By the quadratic formula, the nature of its roots depends entirely on the discriminant : 4. Case 1: Distinct Real Roots — Two Exponentials When , the quadratic formula hands back two different real roots . Each gives an independent solution e^ r_1x and e^ r_2x , so the general solution is their linear combination: Characteristic equation: r^2 - 3r + 2 = 0 General solution: y = c_1e^ x + c_2e^ 2x 5. Case 2: Repeated Root — Decay With an Extra Factor of x When , the quadratic formula collapses to a single value r (a root of multiplicity 2). But a second-order equation needs two independent solutions, and e^ rx alone is only one. The second comes from multiplying by x : Characteristic equation: r^2 - 4r + 4 = 0 General solution: y = c_1e^ 2x + c_2xe^ 2x = (c_1 + c_2x)e^ 2x 6. Case 3: Complex Conjugate Roots — Oscillation

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