082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

560 Chapter 19 Ordinary differential equations I
Substitution into the l.h.s. of Equation (19.9) gives
(−Acosx−Bsinx)+(Acosx+Bsinx+x)
which equals x, and so the complementary function plus the particular integral is
indeed a solution ofthe inhomogeneous equation, as required by Property 2.
19.6 CONSTANTCOEFFICIENTEQUATIONS
Wenowproceedtostudyindetailthosesecond-orderlinearequationswhichhaveconstant
coefficients. The general formof such an equation is
a d2 y
dx +bdy +cy = f(x) (19.10)
2 dx
wherea,b,careconstants. The homogeneous formof Equation (19.10) is
a d2 y
dx +bdy +cy=0 (19.11)
2 dx
Equationsofthisformariseintheanalysisofcircuits.Considerthefollowingexample.
Engineeringapplication19.5
TheLCRcircuit
WritedownthedifferentialequationgoverningthecurrentflowingintheseriesLCR
circuitshown inFigure 19.5.
Solution
UsingKirchhoff’s voltage lawand the individual laws foreachcomponent wefind
L di ∫
dt +iR+1 idt = v(t)
C
If thisequation isnow differentiated w.r.t.t wefind
L d2 i
dt +Rdi 2 dt + 1 C i = dv(t)
dt
Thisisaninhomogeneoussecond-orderdifferentialequation,withtheinhomogeneity
arising from the term dv . When the circuit componentsL,RandC are constants
dt
wehavewhatistermedalineartime-invariantsystem,andthedifferentialequation
then has constant coefficients.
L C R
i
y (t)
Figure19.5
AnLCRcircuit.