Homogeneous second-order linear ordinary differential equation with constant coefficients
Contents |
Equation
Governing equation:
(1) |
where and are real constants.
Solution
Let and be the roots of the characteristic equation
(2) |
Then
(i) If real, then | (3) | |
(ii) If , then | ||
(iii) If , (), then |
References: Bubenik 1997.
Derivation
The solution is expected in the form of and the goal is to determine .
Express derivatives of y:
Substituting back to the original equation yields:
Since , then the following characteristic equation must be satisfied:
The solution of the quadratic characteristic equation is:
Two Real Roots
If , then the characteristic equation has two real roots and the solution is:
One Real Root
If , then the characteristic equation has a single real root and the solution is:
No Real Root
If , 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
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 |