Homogeneous second-order linear ordinary differential equation with constant coefficients
where and are real constants.
Let and be the roots of the characteristic equation
|(i) If real, then||(3)|
|(ii) If , then|
|(iii) If , (), then|
References: Bubenik 1997.
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.
Usage in Structural Engineering
- Second order constant coefficient linear differential equation at EqWorld
- Homogeneous linear equations with constant coefficients at SOS Math
- F. Bubenik, M. Pultar, I. Pultarova: Matematicke vzorce a metody, Vydavatestvi CVUT, Prague 1997, p. 224 (in Czech)
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Method of undetermined coefficients · Method of variation of parameters