Higher-Order Summary

Differential Equations · Axiom Academy

SUMMARY Higher-Order Linear Differential Equations Review how the characteristic equation, undetermined coefficients, and variation of parameters combine to solve any constant-coefficient linear DE — and where they show up in springs and circuits. Every general solution splits as y = y_h + y_p : the homogeneous solution y_h (from the characteristic equation) plus any particular solution y_p . The characteristic equation's roots come in exactly three flavors — distinct real, repeated real, or complex conjugate — and each flavor has its own solution form. Undetermined coefficients is the fast method when g(x) is polynomial, exponential, sine/cosine, or a product of these — but it needs the resonance fix ( ) when the trial form duplicates y_h . Variation of parameters is the universal fallback: it solves for y_p for any continuous g(x) , using the Wronskian of the homogeneous solutions. The Wronskian W = y_1y_2' - y_1'y_2 isn't just bookkeeping — it's the independence test that guarantees y_1, y_2 span the full solution space, and it appears directly in the variation-of-parameters formulas. Core Concept General Solution Structure A linear equation obeys superposition: its general solution is the homogeneous solution y_h (solving the equation with g(x)=0 ) plus any one particular solution y_p . When to use: the starting framework for every higher-order linear DE, constant-coefficient or not.

This is the written version of the interactive lesson above. See the full Differential Equations course.