--- title: "Second-Order Linear Equations with Constant Coefficients" description: "Try an exponential and the equation becomes a quadratic in the exponent. Real, repeated and complex roots give three shapes of general solution." canonical: https://lightmysky.com/learn/mathematics/second-order-linear-equations-with-constant-coefficients-mt_G7sMq1EHwQ source: https://lightmysky.com/learn/mathematics/second-order-linear-equations-with-constant-coefficients-mt_G7sMq1EHwQ.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Second-Order Linear Equations with Constant Coefficients Try an exponential and the equation becomes a quadratic in the exponent. Real, repeated and complex roots give three shapes of general solution. Subject: Mathematics · Area: Differential Equations · Ages 20 to 21 Page: https://lightmysky.com/learn/mathematics/second-order-linear-equations-with-constant-coefficients-mt_G7sMq1EHwQ ## Ready when they can - Write and solve the characteristic equation - Give the general solution in each of the three root cases - Convert a complex-root solution into a decaying sine and cosine form ## Lesson: One guess that cracks constant equations Constant coefficient equations give way to one guess: try y equals e to the r x and see which r work. Substitution turns the differential equation into the characteristic polynomial, a plain quadratic whose roots decide the shape of the solution. The general solution combines two independent pieces with constants C1 and C2, and your starting position plus starting velocity fix those constants. Newton is why you care: springs, circuits, and pendulums all lead here. **Example.** Two distinct real roots give two exponentials, side by side. A repeated root adds a factor of x, because the second piece must stay independent of the first. A complex pair becomes an oscillation wrapped in an exponential, which you rewrite with sine and cosine. Take y double prime plus 4 y prime plus 4 y equals 0: the polynomial is r squared plus 4 r plus 4, which is r plus 2 squared, so r is minus 2 twice and the answer is C1 plus C2 times x, all times e to the minus 2 x. Distinct real roots r1 and r2 give y = C1 e^(r1 x) + C2 e^(r2 x). A repeated root r gives y = (C1 + C2 x) e^(rx). A complex pair a ± bi gives, by Euler's formula, e^(ax)(cos bx + i sin bx), and the real solutions are y = e^(ax)(C1 cos bx + C2 sin bx). The discriminant of the characteristic quadratic picks the case: positive, zero or negative. When a is negative the oscillation decays. **Example.** Fit the constants. Solve y'' - y' - 6y = 0 with y(0) = 1 and y'(0) = 8. The characteristic equation r² - r - 6 = 0 factorises as (r - 3)(r + 2) = 0, so y = C1 e^(3x) + C2 e^(-2x). Position: y(0) = C1 + C2 = 1. Velocity: y' = 3C1 e^(3x) - 2C2 e^(-2x), so y'(0) = 3C1 - 2C2 = 8. Solving, C1 = 2 and C2 = -1, so y = 2e^(3x) - e^(-2x). Check the template before fitting: the wrong template can never meet both conditions. **Recap.** Guess the exponential, read the roots, pick the matching template, then fit the two constants. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [The Logistic Equation and Saturating Growth](https://lightmysky.com/learn/mathematics/the-logistic-equation-and-saturating-growth-mt_7Y_48Ga33L) - [Euler's Formula and the Exponential Form](https://lightmysky.com/learn/mathematics/eulers-formula-and-the-exponential-form-mt_RX2IxsnSwG) - [The Discriminant and the Number of Roots](https://lightmysky.com/learn/mathematics/the-discriminant-and-the-number-of-roots-mt_uj95V0aLVS) ## Opens up - [Solving Initial Value Problems with Laplace Transforms](https://lightmysky.com/learn/mathematics/solving-initial-value-problems-with-laplace-transforms-mt_0BhjPVNUwl) - [Nonhomogeneous Equations and Particular Solutions](https://lightmysky.com/learn/mathematics/nonhomogeneous-equations-and-particular-solutions-mt_gbkQZte_JZ) - [Sturm-Liouville Problems and Eigenfunction Expansions](https://lightmysky.com/learn/mathematics/sturm-liouville-problems-and-eigenfunction-expansions-mt_oqadAPaSsW) - [Recurrence Relations and Characteristic Roots](https://lightmysky.com/learn/mathematics/recurrence-relations-and-characteristic-roots-mt_q0b2bi0iAB) - [Separation of Variables and the Heat Equation](https://lightmysky.com/learn/mathematics/separation-of-variables-and-the-heat-equation-mt_XnPDL0X1TL)