Homogeneous second-order linear ordinary differential equation with constant coefficients

Equation

Governing equation:

$y''+ay'+by=0$

(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 $\lambda _{1}\neq \lambda _{2}$ 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 $\lambda _{1}=\alpha +i\beta$, $\lambda _{1}=\alpha -i\beta$ ($\beta \neq 0$), 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:

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

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

Since $e^{\lambda x}\neq 0$, 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:

$y=C_{1}e^{\lambda _{1}x}+C_{2}xe^{\lambda _{1}x}$

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)

 

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