Second Order Differential Equation

Second Order Differential Equation

An ordinary differential equation of order 2. That is, a differential equation in which the highest derivative is a second derivative.

Example equation: d²y/dx² − dy/dx − 2y = e^x

Key Formula

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

Where:

$y$ = The unknown function of x
$y'$ = The first derivative of y with respect to x
$y''$ = The second derivative of y with respect to x
$a, b, c$ = Constant coefficients (with a ≠ 0)
$f(x)$ = A known function of x; if f(x) = 0 the equation is called homogeneous

Worked Example

Problem: Solve the homogeneous second order differential equation y'' - 5y' + 6y = 0.

Step 1: Assume a solution of the form y = e^{rx} and substitute into the equation. Each derivative brings down a factor of r.

r^2 e^{rx} - 5r\,e^{rx} + 6\,e^{rx} = 0

Step 2: Factor out e^{rx} (which is never zero) to obtain the characteristic equation.

r^2 - 5r + 6 = 0

Step 3: Factor the characteristic equation to find the roots.

(r - 2)(r - 3) = 0 \implies r = 2 \text{ or } r = 3

Step 4: Because the two roots are real and distinct, the general solution is a linear combination of the two corresponding exponentials.

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.

Another Example

This example differs from the first by featuring complex roots of the characteristic equation, which produce a trigonometric solution, and by including initial conditions that pin down the specific constants.

Problem: Solve the second order differential equation y'' + 4y = 0 with initial conditions y(0) = 3 and y'(0) = -2.

Step 1: Write the characteristic equation by substituting y = e^{rx}.

r^2 + 4 = 0 \implies r = \pm\,2i

Step 2: The roots are complex: r = 0 ± 2i. For complex roots α ± βi, the general solution uses sines and cosines. Here α = 0 and β = 2.

y = C_1\cos(2x) + C_2\sin(2x)

Step 3: Apply the first initial condition y(0) = 3.

y(0) = C_1\cos(0) + C_2\sin(0) = C_1 = 3

Step 4: Differentiate and apply the second initial condition y'(0) = -2.

y' = -2C_1\sin(2x) + 2C_2\cos(2x) \implies y'(0) = 2C_2 = -2 \implies C_2 = -1

Answer: The particular solution is

y = 3\cos(2x) - \sin(2x)

Frequently Asked Questions

What is the difference between a first order and a second order differential equation?

A first order differential equation contains at most the first derivative y', while a second order differential equation contains the second derivative y'' as its highest derivative. Because second order equations have two independent solutions, their general solution involves two arbitrary constants instead of one.

How do you solve a second order linear differential equation with constant coefficients?

You substitute y = e^{rx} into the equation to produce a characteristic (auxiliary) equation in r. If the roots are real and distinct, the solution uses two exponentials. If the roots are repeated, you multiply one term by x. If the roots are complex, the solution involves sines and cosines. For non-homogeneous equations, you also find a particular solution using undetermined coefficients or variation of parameters.

Why does a second order differential equation have two constants in its general solution?

The order of a differential equation determines the number of integrations needed to recover the original function. Each integration introduces one arbitrary constant. Since a second order equation requires two integrations, its general solution contains exactly two arbitrary constants, typically determined by two initial or boundary conditions.

Second Order Differential Equation vs. First Order Differential Equation

	Second Order Differential Equation	First Order Differential Equation
Highest derivative	y'' (second derivative)	y' (first derivative)
General solution constants	Two arbitrary constants (C₁, C₂)	One arbitrary constant (C)
Characteristic equation	Quadratic in r	Linear in r (for linear constant-coefficient type)
Typical solution method	Characteristic equation, undetermined coefficients, variation of parameters	Separation of variables, integrating factor
Conditions needed for unique solution	Two initial/boundary conditions	One initial condition

Why It Matters

Second order differential equations appear throughout physics and engineering. Newton's second law, F = ma, is itself a second order ODE because acceleration is the second derivative of position. You encounter these equations when studying spring-mass systems, electric circuits (RLC circuits), wave motion, and beam deflection, making them essential in any calculus or differential equations course.

Common Mistakes

Mistake: Forgetting the extra x-factor when the characteristic equation has a repeated root.

Correction: When the characteristic equation gives a repeated root r = r₁, the two independent solutions are e^{r₁x} and x·e^{r₁x}. Without the factor of x, you would only have one independent solution, which is not a complete general solution.

Mistake: Confusing the signs when reading off roots of the characteristic equation, especially with complex roots.

Correction: For r² + 4 = 0, the roots are r = ±2i, not r = ±2. Always check whether the discriminant b² − 4ac is negative before deciding the root type. Complex roots α ± βi lead to solutions involving e^{αx}cos(βx) and e^{αx}sin(βx).

Related Terms

Ordinary Differential Equation — The broader class that includes second order equations
Order of a Differential Equation — Defines the order based on the highest derivative
Differential Equation — General term for equations involving derivatives
Derivative — Fundamental operation appearing in every differential equation
Second Derivative — The specific derivative that defines second order