Second Order Ordinary Differential Equation — Definition, Formula & Examples

A second order ordinary differential equation (ODE) is an equation that contains the second derivative of an unknown function with respect to a single independent variable, and no higher derivatives. It describes how a quantity's rate of change itself changes, which is why it appears naturally in problems involving acceleration, curvature, and oscillation.

A second order ODE is an equation of the form $F(x, y, y', y'') = 0$ , where $y = y(x)$ is the unknown function of the single independent variable $x$ , $y' = dy/dx$ , and $y'' = d^2y/dx^2$ . The equation is called ordinary because $y$ depends on only one independent variable, and second order because $y''$ is the highest-order derivative present.

Key Formula

a\,y'' + b\,y' + c\,y = g(x)

Where:

$y$ = Unknown function of x
$y', y''$ = First and second derivatives of y with respect to x
$a, b, c$ = Constant (or variable) coefficients, with a ≠ 0
$g(x)$ = Forcing function; g(x) = 0 gives the homogeneous case

How It Works

To solve a second order ODE, you typically need two initial or boundary conditions because the general solution contains two arbitrary constants. For linear equations with constant coefficients, such as

ay'' + by' + cy = 0

, you substitute

y = e^{rx}

to obtain a characteristic equation

ar^2 + br + c = 0

. The nature of the roots — real and distinct, repeated, or complex — determines whether the general solution involves exponentials, a polynomial factor, or sines and cosines. Non-homogeneous equations (right-hand side not zero) require an additional particular solution found via undetermined coefficients or variation of parameters.

Worked Example

Problem: Solve the second order ODE y'' − 5y' + 6y = 0.

Step 1: Assume a solution of the form y = e^(rx) and substitute into the equation to get the characteristic equation.

r^2 - 5r + 6 = 0

Step 2: Factor the characteristic equation to find the roots.

(r - 2)(r - 3) = 0 \implies r = 2,\; r = 3

Step 3: Since the roots are real and distinct, the general solution is a linear combination of the two exponential solutions.

y = C_1 e^{2x} + C_2 e^{3x}

Answer: The general solution is

y = C_1 e^{2x} + C_2 e^{3x}

, where

C_1

and

C_2

are arbitrary constants determined by initial or boundary conditions.

Why It Matters

Newton's second law

F = ma

is itself a second order ODE, making this equation type central to mechanics, electrical circuit analysis, and vibration engineering. Mastering these equations is essential in any introductory ODE or engineering mathematics course.

Common Mistakes

Mistake: Writing only one arbitrary constant in the general solution instead of two.

Correction: A second order ODE always requires two independent constants in its general solution, corresponding to the two initial/boundary conditions needed to specify a unique solution.