Skip to content

Home Learn Practice About

Sign in Create account

Math Infinitum

Mapping the lesson.

Loading the workspace…

Mathematics, made tangible.

Explore

Home
Learn
Practice
About

Elsewhere

YouTube
Instagram
TikTok
Ko-fi
Telegram
Twitter

© 2026 Math InfinitumPrivacy Policy Development / Legacy prototype

Second-order linear equations · Differential Equations | Math Infinitum

Learn
Differential Equations
Second-order linear equations

Differential Equations · Chapter 04

Second-order linear equations

Equations in the second derivative, which a value and a slope determine. Solutions add and scale, so the whole answer is a two-dimensional space; for constant coefficients an exponential guess turns the equation into a quadratic, and its roots — real, repeated or complex — say what every solution does.

Equations in the second derivative, which a value and a slope determine. Solutions add and scale, so the whole answer is a two-dimensional space; for constant coefficients an exponential guess turns the equation into a quadratic, and its roots — real, repeated or complex — say what every solution does.

a y^{''} + b y^{'} + cy = 0

a r^{2} + b r + c = 0

y = C_{1} y_{1} + C_{2} y_{2}

e^{r t}

W (y_{1}, y_{2})

y (t_{0}) = y_{0}

y^{'} (t_{0}) = v_{0}

e^{α t} cos β t

Lessons

Build the foundation

01Second-order linear equationsNewton's law puts a second derivative into physics: force is mass times acceleration, and acceleration is the second derivative of position. An equation in the second derivative remembers more than a first-order one — not only where the solution is, but which way it is heading — and so it needs more to pin a solution down: a value and a slope at the starting point. This chapter is about the linear ones, where that pair of numbers always picks exactly one solution.Read lesson →
02SuperpositionFor a homogeneous linear equation, solutions can be added and multiplied by constants and they remain solutions. This is superposition, and it is why linear equations are tractable: find two independent solutions and every combination of them is available. It fails the moment the equation is forced or nonlinear, and the ways it fails are as instructive as the principle.Read lesson →
03The characteristic equationFor an equation with constant coefficients there is a natural guess: an exponential, because differentiating an exponential only multiplies it by its rate. Substituting it turns the differential equation into an algebraic one, a quadratic for the rate, and the next three lessons are about what its roots say.Read lesson →
04Two distinct real rootsWhen the characteristic equation has two different real roots, the equation has two exponential solutions, one growing or decaying at each rate, and every solution is a combination of them. The long run is read off the signs: two negative roots mean every solution dies away, a positive root means almost every solution grows, and the larger root wins in the end.

Read lesson →

05Repeated rootsWhen the characteristic equation has a repeated root, the exponential guess gives only one solution, and a second-order equation needs two. The second is the same exponential multiplied by x. It looks like a trick at first and is nothing of the kind: at a repeated root both the polynomial and its derivative vanish, and those are exactly the two things the new solution needs.Read lesson →

06Complex rootsWhen the discriminant is negative the roots are a complex pair, and their exponentials are complex-valued. Euler's formula rewrites them, and their real and imaginary parts are two real solutions: an exponential times a cosine, and the same exponential times a sine. The imaginary part of the root is a frequency, and the real part decides whether the oscillation grows, holds or dies.Read lesson →

07Initial conditionsWith the general solution in hand, a value and a slope at the starting point give two linear equations for the two constants. Solving them is a small piece of linear algebra, and it always works — the next lesson explains why the system is never singular. What does not always work is asking for values at two different points: that is a boundary value problem, and it can have one solution, none, or infinitely many.Read lesson →

08The Wronskian and independence of solutionsTwo solutions give every solution only if they are genuinely different — linearly independent, in the language of the linear algebra course. The test is a determinant: the Wronskian, built from the two solutions and their derivatives, which is the determinant of the matrix of the last lesson. If it is not zero at one point, the two solutions form a basis of the solution space, which therefore has dimension two.Read lesson →

09Chapter practiceBring the chapter together: write the characteristic equation, tell which kind of roots it has, build the general solution, fit it to a value and a slope, and test a pair of solutions with the Wronskian. No worked example sits above the answers.Read lesson →