Converting y'' + 3y' + 2y = 0

Differential Equations · Axiom Academy

EXAMPLE Converting y'' + 3y' + 2y = 0 to a System Learn to transform a second-order differential equation into an equivalent first-order system. Convert the second-order linear differential equation y'' + 3y' + 2y = 0 into an equivalent first-order system of the form . Excellent work! You've converted a second-order differential equation into a first-order system. Here's what we learned: State variables: setting x_1 = y and x_2 = y' captures both the function and its derivative as two first-order unknowns. Two equations: the first equation x_1' = x_2 comes straight from the definitions. The second, x_2' = -2x_1 - 3x_2 , comes from solving the original ODE for y'' . Matrix form: the system is with companion matrix — the bottom row is the original equation's coefficients, negated and reversed in sign, in coefficient order. Equivalence: the system's eigenvalues, -1 and -2 , are exactly the roots of the scalar characteristic equation r^2 + 3r + 2 = 0 — the two forms describe the same solutions. This technique extends to any higher-order linear equation and is fundamental for numerical methods, stability analysis, and phase-plane analysis.

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