Homogeneous second-order linear ordinary differential equation with constant coefficients

Equation

Governing equation:

$y'' + a y' + b y = 0$

(1)

where $a$ and $b$ are real constants.

Solution

Let $\lambda_1$ and $\lambda_2$ be the roots of the characteristic equation

$f(\lambda) = \lambda^2 + a \lambda + b = 0$

(2)

Then

	(i) If $\lambda_1 \ne \lambda_2$ real, then $y = C_1 e^{\lambda_1 x} + C_2 e^{\lambda_1 x}$	(3)
	(ii) If $\lambda = \lambda_1 = \lambda_2$ , then $y = (C_1 x + C_2) e^{\lambda x}$
	(iii) If $\lambda_1 = \alpha + i \beta$ , $\lambda_1 = \alpha - i \beta$ ( $\beta \ne 0$ ), then $y = e^{\alpha x} (C_1 \cos(\beta x) + C_2 \sin (\beta x)) = e^{\alpha x} C \cos(\beta x + \theta)$

References: Bubenik 1997.

Derivation

The solution is expected in the form of $y = e^{\lambda x}$ and the goal is to determine $\lambda$ .

Express derivatives of y:

$y = e^{\lambda x}$

$y' = \lambda e^{\lambda x}$

$y'' = \lambda^2 e^{\lambda x}$

Substituting back to the original equation yields:

$\lambda^2 e^{\lambda x} + a \lambda e^{\lambda x} + b e^{\lambda x} = 0$

$e^{\lambda x} (\lambda^2 + a \lambda + b) = 0$

Since $e^{\lambda x} \ne 0$ , then the following characteristic equation must be satisfied:

$\lambda^2 + a \lambda + b = 0$

The solution of the quadratic characteristic equation is:

$\lambda_{1,2} = \frac{-a \pm \sqrt{a^2-4b}}{2}$

Two Real Roots

If $\sqrt{a^2-4b} > 0$ , then the characteristic equation has two real roots and the solution is:

$y = C_1 e^{\lambda_1 x} + C_2 e^{\lambda_2 x}$

One Real Root

If $\sqrt{a^2-4b} = 0$ , then the characteristic equation has a single real root and the solution is:

$y = C_1 e^{\lambda_1 x} + C_2 x e^{\lambda_1 x}$

No Real Root

If $\sqrt{a^2-4b} < 0$ , then the characteristic equation has a two complex roots.

To Do: derivation of solution for second order linear ordinary differential equation - clean up stuff. (who: user:ok; priority: 100; hours: 0) (all) (cat)

Usage in Structural Engineering

Single degree of freedom damped free vibrations

External Links

Second order constant coefficient linear differential equation at EqWorld
Homogeneous linear equations with constant coefficients at SOS Math

References

F. Bubenik, M. Pultar, I. Pultarova: Matematicke vzorce a metody, Vydavatestvi CVUT, Prague 1997, p. 224 (in Czech)

Home > Topics > Mathematics > Differential Equations e
Overview	Differential Equations
Ordinary differential equations	Homogeneous second-order linear ordinary differential equation with constant coefficients Nonhomogeneous second-order linear ordinary differential equation with constant coefficients Method of undetermined coefficients · Method of variation of parameters

Homogeneous second-order linear ordinary differential equation with constant coefficients

Contents

Equation

Solution

Derivation

Two Real Roots

One Real Root

No Real Root

Usage in Structural Engineering

External Links

References

Personal tools

Namespaces

Variants

Views

Actions

Search

Navigation

external links

administration

Toolbox