second order linear differential equation with constant coefficients

Consider the second order homogeneous linear differential equation

x′′+bx+cx=0, (1)

where b and c are real constants.

The explicit solution is easily found using the characteristic equationMathworldPlanetmathPlanetmathPlanetmath method. This method, introduced by Euler, consists in seeking solutions of the form x(t)=ert for (1). Assuming a solution of this form, and substituting it into (1) gives



r2+br+c=0 (2)

which is called the characteristic equation of (1). Depending on the nature of the roots ( r1 and r2 of (2), there are three cases.

  • If the roots are real and distinct, then two linearly independentMathworldPlanetmath solutions of (1) are

  • If the roots are real and equal, then two linearly independent solutions of (1) are

  • If the roots are complex conjugatesMathworldPlanetmath of the form r1,2=α±iβ, then two linearly independent solutions of (1) are


The general solution to (1) is then constructed from these linearly independent solutions, as

ϕ(t)=C1x1(t)+C2x2(t). (3)

Characterizing the behavior of (3) can be accomplished by studying the two-dimensional linear system obtained from (1) by defining y=x:

x =y (4)
y =-by-cx. (5)

Remark that the roots of (2) are the eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the Jacobian matrix of (5). This generalizes to the characteristic equation of a differential equationMathworldPlanetmath of order n and the n-dimensional system associated to it.

Also note that the only equilibrium of (5) is the origin (0,0). Suppose that c0. Then (0,0) is called a

roman]enumerate source iffb<0andc>0, spiral source iffitisasourceandb2-4c<0, sink iffb>0andc>0, spiral sink iffitisasinkandb2-4c<0, iffc<0, center iffb=0andc>0.Titlesecond order linear differential equation with constant coefficientsCanonical nameSecondOrderLinearDifferentialEquationWithConstantCoefficientsDate of creation2013-03-22 13:24:49Last modified on2013-03-22 13:24:49OwnerMathprof (13753)Last modified byMathprof (13753)Numerical id9AuthorMathprof (13753)Entry typeTopicClassificationmsc 34A30Classificationmsc 34-01Classificationmsc 34C05Related topicGeneralSolutionOfLinearDifferentialEquationRelated topicTelegraphEquationDefinescharacteristic equationDefinessourceDefinessinkDefinescenter