What does linear mean?
A differential equation is a rule connecting a function to its own rates of change. The unknown is not a number — it is an entire curve y(x). The equation tells you how the function's value, its slope, and its curvature must relate to each other at every point simultaneously.
An equation is linear when y and all its derivatives appear only to the first power, are never multiplied together, and are never fed into functions like sin or eʸ. The coefficients can depend freely on x — that is fine.
Linearity gives you a superpower: if y₁ and y₂ are both solutions to a homogeneous linear ODE, then c₁y₁ + c₂y₂ is also a solution for any constants c₁, c₂. You can add and scale solutions freely. This is the structural backbone of everything that follows.
Click any equation to see why it is or is not linear.
Why g(x) = 0 means no external driving
Think of the left side as a machine — the operator L — that processes the function y through its own internal rule. The right side g(x) is entirely separate: it does not involve y at all. It is a prescribed function sitting there, pushed in from outside.
The equation Ly = g(x) says: the system's internal dynamics, acting on y, must match this external input at every moment. When g(x) = 0, the external prescription vanishes. The equation becomes Ly = 0, meaning the system is in balance with itself — no outside agent is present. This is the natural response.
Ly = 0
No external term. The system evolves entirely under its own internal rules. g(x) = 0 is not just zero on the right — it means no external function exists in the equation at all.
Ly = g(x)
An external function is present, driving the system. Without g(x) the equation could never produce this output on its own. The solution gains a forced component on top of the natural one.
Solution space as a vector space
A second-order ODE involves y″, which means two integrations are needed to recover y. Each introduces one free constant. The general solution must therefore contain exactly two arbitrary constants — you need two independent building blocks.
This is not a coincidence. The set of all solutions to a homogeneous linear ODE forms a vector space — the same structure as the null space {x : Ax = 0} in matrix algebra. Just as a null space needs a basis, the solution space needs linearly independent solutions.
Ax = 0
Matrix operator on a vector. Solutions form the null space, spanned by n basis vectors.
Ly = 0
Differential operator on a function. Solutions form a function space, spanned by n independent solutions.
y = c₁y₁ + c₂y₂
The general solution is any linear combination of the basis functions — identical pattern to vectors.
This is not an analogy — it is the same mathematics. Functions are vectors in an infinite-dimensional space. Differential operators are linear maps. The entire theory of linear ODEs is linear algebra with functions instead of arrows.
The Wronskian
Two functions are linearly independent if one is not just a constant multiple of the other. If they are dependent you effectively have one direction in the solution space, not two — and you cannot span the full general solution.
The Wronskian is defined as the determinant of a matrix built from the two candidate solutions and their derivatives:
The geometric intuition: in 2D linear algebra, the determinant of two column vectors measures the area of the parallelogram they span. If that area is zero, the vectors point in the same direction — they are dependent. The Wronskian is the exact same test applied to functions — treat each function as a "direction" in solution space. If W ≠ 0 at any point on the interval, the functions are independent and form a valid basis.
Select a pair of functions to compute the Wronskian step by step.
Why try y = erx?
For constant-coefficient equations, we need a function that survives differentiation intact — one where taking the derivative just multiplies by a constant without changing the shape. Exponentials are the only such class:
Every derivative is still eʳˣ, just scaled by a power of r. When you substitute into ay″ + by′ + cy = 0, every term shares the factor eʳˣ:
Since eʳˣ is never zero, divide it out. The entire differential equation collapses to a quadratic in r — the characteristic equation. Calculus becomes algebra.
Watch what happens when you differentiate eʳˣ repeatedly. Drag the slider to change r.
The three root cases
Once you solve ar² + br + c = 0, the discriminant b² − 4ac tells you what kind of roots you have. Each type produces a fundamentally different solution shape — and a different physical behavior.
When b² − 4ac > 0, the two roots r₁ ≠ r₂ are both real. Each gives an independent exponential solution:
The physical behavior is pure exponential growth or decay (or a combination), depending on the signs of the roots. No oscillation.
Solve y″ + by′ + cy = 0. Drag sliders to change b and c and see the solution shape update in real time.
When b² − 4ac = 0, there is one repeated root r. A single eʳˣ gives only one solution — a scaled copy of itself is dependent. The second independent solution is xeʳˣ.
You can verify xeʳˣ is a solution: substitute it back in and every term cancels. The x factor provides the independence without changing the fundamental exponential character.
When b² − 4ac < 0, the roots are a complex conjugate pair α ± βi. Euler's formula converts the complex exponentials to real-valued solutions:
The eᵅˣ factor is the envelope — it controls growth (α > 0), decay (α < 0), or stays at 1 (α = 0). The cos and sin provide the oscillation at frequency β.
Initial Value Problems
The general solution contains two free constants c₁ and c₂. Initial conditions pin down which specific solution matches the physical situation. You need exactly two: one for y and one for y′, both evaluated at a starting point x₀.
The workflow: find the general solution → differentiate it → evaluate both at x₀ → solve the 2×2 linear system for c₁ and c₂.
Enter the equation y″ + by′ + cy = 0 and initial conditions. Every step is shown.
Everything connected
Every concept in this guide is one part of a single coherent machine. The theory explains the why; the characteristic equation method delivers the how.
| Concept | What it answers | Key test |
|---|---|---|
| Linear ODE | Can solutions be freely added and scaled? | y, y′ appear to first power only, not multiplied |
| Homogeneous (g = 0) | Is there any external driving present? | Right-hand side is identically zero |
| n independent solutions | How many basis functions does the solution space need? | Equals the order of the ODE |
| Wronskian | Are two candidate solutions truly independent? | W = y₁y₂′ − y₂y₁′ ≠ 0 |
| Characteristic equation | What r makes eʳˣ a solution? | ar² + br + c = 0 |
| Three root cases | What shape does the solution take? | Sign of discriminant b² − 4ac |
| Initial conditions | Which specific solution matches this physical state? | Solve 2×2 system for c₁, c₂ |
Linearity gives superposition. Homogeneity means no forcing. The solution space is a vector space requiring n basis functions. The Wronskian tests independence of candidates. The characteristic equation finds those candidates efficiently for constant coefficients. Initial conditions select one from the family. Each idea feeds the next — none are optional.