Second-Order Linear ODEs

§ 01The Standard Form

A second-order linear ODE with constant coefficients has the form ay'' + by' + cy = g(x), where a, b, c are real constants and a ≠ 0. When g(x) = 0, the equation is homogeneous, and that is exactly what this page covers.

The requirement that the coefficients a, b, c are constants (not functions of x) is what makes this family of ODEs particularly tractable. Variable-coefficient second-order ODEs (like Euler's equation or Bessel's equation) require different, more advanced techniques.

The equation is second order, so its general solution contains exactly two arbitrary constants, C₁ and C₂. To determine both constants, you need two initial conditions: typically the value of y and the value of y' at some point x = x₀.

Why Solutions Have the Form y = eʳˣ

The key insight that unlocks this entire family of equations is to guess that a solution has the exponential form y = eʳˣ for some constant r. This is not a wild guess, exponential functions are the only functions that are proportional to all of their own derivatives, which is exactly what the equation ay'' + by' + cy = 0 requires.

If y = eʳˣ, then y' = reʳˣ and y'' = r²eʳˣ. Substituting into the ODE:

This quadratic in r is called the characteristic equation. Its roots determine everything about the solution. There are exactly three cases, depending on the discriminant b² − 4ac.

§ 02The Three Root Cases at a Glance

Solve ar² + br + c = 0 and check the discriminant. The nature of the roots completely determines the form of the general solution.

§ 03Case 1 — Two Distinct Real Roots

When b² − 4ac > 0, the quadratic ar² + br + c = 0 has two distinct real roots r₁ and r₂. Both y = e^r₁x and y = e^r₂x are solutions, and because they are linearly independent, their linear combination is the general solution.

Two functions are linearly independent if neither is a constant multiple of the other. For distinct exponentials e^r₁x and e^r₂x with r₁ ≠ r₂, this is always true, their Wronskian W = (r₂ − r₁)e^(r₁+r₂)x is never zero.

Physical Interpretation: Overdamped Systems

In a spring-mass-damper system, this case corresponds to overdamping, the damping force is so strong that the system returns to equilibrium without oscillating. Both exponential terms either decay (if r₁, r₂ < 0) or grow (if positive), but there is no oscillation.

§ 04Case 2 — Repeated Real Root

When b² − 4ac = 0, the quadratic has a single real root r = −b/(2a) of multiplicity two. Using e^rx twice gives only one independent solution, so we multiply by x to generate the second.

Why Does xe^rx Work?

If r is a double root of ar² + br + c = 0, then we can verify directly by substituting y₂ = xe^rx:

y₂' = e^rx + rxe^rx = (1 + rx)e^rx

y₂'' = re^rx + r(1 + rx)e^rx = (2r + r²x)e^rx

Substituting into ay'' + by' + cy and collecting terms yields a(r² + br + c)xe^rx + (2ar + b)e^rx. Because r is a double root, ar² + br + c = 0 and 2ar + b = 0 (the derivative of the characteristic polynomial at a double root is also zero), so the whole expression equals zero. Therefore xe^rx is indeed a solution.

The two solutions e^rx and xe^rx are linearly independent (their Wronskian equals e^2rx ≠ 0), so the general solution is their linear combination.

Physical Interpretation: Critical Damping

In mechanics, a repeated root corresponds to critical damping, the boundary between oscillatory (underdamped) and purely exponential (overdamped) behaviour. A critically damped system returns to equilibrium as quickly as possible without oscillating. This is the ideal for car suspension systems and door closers.

§ 05Case 3 — Complex Conjugate Roots

When b² − 4ac < 0, the characteristic equation has no real roots. Instead it has a pair of complex conjugate roots r = α ± βi, where α = −b/(2a) and β = √(4ac − b²)/(2a). The resulting real general solution involves oscillations wrapped in an exponential envelope.

Where Does This Form Come From? — Euler's Formula

The characteristic equation gives roots r₁ = α + βi and r₂ = α − βi, so the complex-valued general solution is y = Ae^(α+βi)x + Be^(α−βi)x. Using Euler's formula e^iθ = cos θ + i sin θ, we can write:

Taking real linear combinations: C₁ = (A + B) and C₂ = i(A − B) produces the real-valued solution above. The constants C₁ and C₂ are real when we impose the requirement of real initial conditions.

The Three Sub-Cases of Oscillation

Physical Interpretation: Underdamped Systems

A mass on a spring with light damping exhibits exactly this behaviour. The oscillation frequency is β/(2π) Hz, and the decay rate is controlled by α. The free oscillation of guitar strings, the ringing of a tuning fork, and the oscillation of an LC electrical circuit are all described by solutions of this form.

§ 06The Wronskian and Linear Independence

Two solutions y₁ and y₂ form a fundamental set of solutions, and hence give the true general solution, if and only if they are linearly independent. The Wronskian is the test.

If W ≠ 0 at any point in the interval of interest, the two solutions are linearly independent and their linear combination C₁y₁ + C₂y₂ is the complete general solution. Abel's theorem tells us something stronger: if W ≠ 0 at one point, then W ≠ 0 everywhere on the interval (for constant-coefficient equations).

For the three cases above:

In all three cases the Wronskian is never zero, confirming that our solution pairs are always linearly independent. This guarantees that every solution of the ODE can be written as C₁y₁ + C₂y₂ for some constants C₁ and C₂: the general solution truly is general.

§ 07Common Mistakes

§ 0810-Question Quiz

Work through every question. Each tests a different aspect of the three-case method.

Continue Learning

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