Higher-Order Constant Coefficient Equations

The methods for second-order linear equations extend naturally to higher-order equations with constant coefficients, providing systematic solution techniques for many physical and engineering problems.

Definitionn-th Order Linear Homogeneous ODE

An n-th order linear homogeneous differential equation with constant coefficients has the form:

$a_ny^{(n)} + a_{n-1}y^{(n-1)} + \cdots + a_1y' + a_0y = 0$

where $a_0, a_1, \ldots, a_n$ are real constants with $a_n \neq 0$ .

Characteristic Equation Method

The solution method parallels the second-order case. Seeking solutions of the form $y = e^{rx}$ leads to the characteristic equation:

$a_nr^n + a_{n-1}r^{n-1} + \cdots + a_1r + a_0 = 0$

This is an n-th degree polynomial equation whose roots determine the general solution.

ExampleThird-Order Equation

Solve $y''' - 6y'' + 11y' - 6y = 0$ .

Characteristic equation:

$r^3 - 6r^2 + 11r - 6 = 0$

Factoring: $(r-1)(r-2)(r-3) = 0$

Roots: $r_1 = 1$ , $r_2 = 2$ , $r_3 = 3$ (all distinct and real)

General solution:

$y = c_1e^x + c_2e^{2x} + c_3e^{3x}$

Cases for Roots

Case 1: Distinct Real Roots

If the characteristic equation has n distinct real roots $r_1, r_2, \ldots, r_n$ , the general solution is:

$y = c_1e^{r_1 x} + c_2e^{r_2 x} + \cdots + c_ne^{r_n x}$

Case 2: Repeated Real Roots

If a root $r$ has multiplicity $m$ , it contributes m linearly independent solutions:

$e^{rx}, \quad xe^{rx}, \quad x^2e^{rx}, \quad \ldots, \quad x^{m-1}e^{rx}$

ExampleRepeated Roots

Solve $y^{(4)} - 4y''' + 6y'' - 4y' + y = 0$ .

Notice this is $(r-1)^4 = 0$ , giving a root $r = 1$ with multiplicity 4.

General solution:

$y = (c_1 + c_2x + c_3x^2 + c_4x^3)e^x$

Case 3: Complex Roots

Complex roots occur in conjugate pairs. If $r = \alpha \pm i\beta$ with multiplicity $m$ , they contribute:

$e^{\alpha x}\cos\beta x, \quad xe^{\alpha x}\cos\beta x, \quad \ldots, \quad x^{m-1}e^{\alpha x}\cos\beta x$

$e^{\alpha x}\sin\beta x, \quad xe^{\alpha x}\sin\beta x, \quad \ldots, \quad x^{m-1}e^{\alpha x}\sin\beta x$

ExampleMixed Roots

Solve $y^{(4)} + 2y'' + y = 0$ .

Characteristic equation: $r^4 + 2r^2 + 1 = (r^2 + 1)^2 = 0$

Roots: $r = \pm i$ (each with multiplicity 2)

So $\alpha = 0$ , $\beta = 1$ , and we have:

$y = (c_1 + c_2x)\cos x + (c_3 + c_4x)\sin x$

Remark

The number of linearly independent solutions always equals the order of the differential equation. For an n-th order equation, the general solution contains exactly n arbitrary constants.

ExampleMechanical Vibrations

A mass-spring system with three masses leads to a third-order equation. The characteristic roots determine whether the system exhibits:

Distinct real roots: Overdamped (no oscillation)
Complex roots: Oscillatory motion with exponential envelope
Repeated roots: Critical damping transitions

The mathematical structure directly reflects the physical behavior.

Higher-order equations with constant coefficients are completely solvable once the characteristic polynomial is factored, though factoring high-degree polynomials may require numerical methods.