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Unformatted text preview: 15-11SECTION 15.2..Nonhomogeneous Equations: Undetermined Coefficients123115.2NONHOMOGENEOUS EQUATIONS:UNDETERMINED COEFFICIENTSImagine yourself trying to videotape an important event. You might be more concernedwith keeping a steady hand than with understanding the mathematics of motion control,but mathematics plays a vital role in helping you produce a professional-looking tape. Insection 15.1, we modeled mechanical vibrations when the motion is started by an initialdisplacement or velocity. In this section, we extend that model to cases where an externalforce such as a shaky hand continues to affect the system.TODAYIN MATHEMATICSShigefumi Mori (1951–)A Japanese mathematician whoearned the Fields Medal in 1990.A colleague wrote, “The mostprofound and excitingdevelopment in algebraicgeometry during the last decadeor so was the Minimal ModelProgram or Mori’s Program....Shigefumi Mori initiated theprogram with a decisively newand powerful technique, guidedthe general research directionwith some good collaboratorsalong the way and finally finishedup the program by himselfovercoming the last difficulty....Mori’s theorems were stunningand beautiful by the totally newfeatures unimaginable by thosewho had been working, probablyvery hard too, only in thetraditional world of algebraic...surfaces.’’The starting place for our model again is Newton’s second law of motion:F=ma. Wenow add an external force to the spring force and damping force considered in section 15.1.If the external force isF(t) andu(t) gives the displacement from equilibrium, as definedbefore, we havemu(t)= −ku(t)−cu(t)+F(t)ormu(t)+cu(t)+ku(t)=F(t).(2.1)The only change from the spring model in section 15.1 is that the right-hand side of theequation is no longer zero. Equations of the form (2.1) with zero on the right-hand side arecalledhomogeneous.In the case whereF(t)=0, we call the equationnonhomogeneous.Our goal is to find the general solution of such equations (that is, the form of all solu-tions). We can do this by first finding oneparticular solutionup(t) of the nonhomogeneousequation (2.1). Notice that ifu(t) is any other solution of (2.1), then we have thatm(u−up)+c(u−up)+k(u−up)=(mu+cu+ku)−(mup+cup+kup)=F(t)−F(t)=.That is, the functionu−upis a solution of the homogeneous equationmu+cu+ku=solved in section 15.1. So, if the general solution of the homogeneous equation isc1u1+c2u2,thenu−up=c1u1+c2u2, for some constantsc1andc2andu=c1u1+c2u2+up.We summarize this in Theorem 2.1.THEOREM 2.1Letu=c1u1+c2u2be the general solution ofmu+cu+ku=0 and letupbe anysolution ofmu+cu+ku=F(t).Then the general solution ofmu+cu+ku=F(t) is given byu=c1u1+c2u2+up.We illustrate this result with example 2.1....
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- Spring '10
- Constant of integration, Shigefumi Mori