Second-Order Linear ODEs with Constant Coefficients

GRE Math Subject · Axiom Academy

LESSON Second-Order Linear ODEs with Constant Coefficients The characteristic equation method -- the single most tested DE technique on the GRE A second-order linear ODE with constant coefficients has the form: where are real constants with . The key idea: guess that solutions have the form . Substituting into the ODE gives: Since , we can divide through to get the characteristic equation : This is the single most important equation in this entire unit. The nature of the roots determines the form of the general solution. When the discriminant 0"> , the characteristic equation has two distinct real roots . GRE-style follow-up: If and , find the particular solution. When , there is one repeated root . Using alone only gives one solution. The second independent solution is (obtained via reduction of order). Why ? The Wronskian , confirming linear independence. This is the standard reduction of order result: when one solution is known, try . Complex Roots and Real Solutions When , the roots are complex conjugates: where and . Using Euler's formula , we extract real-valued solutions: Physical interpretation: The factor controls growth/decay, while and produce oscillation. When , solutions exhibit damped oscillation . When and have the same sign, the roots are purely imaginary: . GRE problem: Find all solutions of . Similarly, recognize (or equivalently, ). Extension to Higher-Order Equations

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