Subject GRE-Style: Characteristic Equation

GRE Math Subject · Axiom Academy

EXAMPLE GRE-Style: Characteristic Equations Four worked problems on second-order ODEs -- try each before revealing the solution What is the general solution of ? Solution: The characteristic equation is: Roots: and (distinct real roots). Speed note: Factor the quadratic mentally. The coefficients suggest factors that multiply to and add to : that is and . Find the solution of satisfying and . Solution: Characteristic equation: Repeated root . General solution: Answer: (A) and (C) are both . On the real GRE only one form would appear. Key takeaway: Recognize perfect squares instantly. screams "repeated root." Then the derivative of requires product rule. The equation has solutions that: Decay exponentially without oscillation Oscillate with constant amplitude Oscillate with decaying amplitude Oscillate with growing amplitude The factor causes decay, while and cause oscillation. Answer: (D) -- oscillate with decaying amplitude. Quick classification: You do not need to compute the roots explicitly. Since , roots are complex. Since the real part is , amplitude decays. Done in 15 seconds. Find the general solution of . Recall that , so the root contributes a constant term. Common trap: Students forget that is a root, giving only 2 terms instead of 3. A third-order ODE must have 3 arbitrary constants. If your answer has fewer, you missed a root. Choice (B) is the classic wrong answer.

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