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

652 Chapter 21 The Laplace transform
The unknown constants, A, B andC, need to be expressed in terms of the known
constantsRandL. Multiplying the above equation bys 2 (R +Ls) yields
1 =As(R +Ls) +B(R +Ls) +Cs 2 (1)
Lettings = 0 inEquation (1)gives
1=BR
andsoB= 1 R .
Wenotethatwhens = − R L thenR+Lsis0.Hencelettings = −R L inEquation(1)
gives
(
1 =C − R ) 2
fromwhichC = L2
L R 2
Comparing the coefficient ofson both sides of Equation (1) gives
0=AR+BL
A = − BL
R = − L R 2
Substituting the expressions forA,BandC into the expression forI(s)gives
I(s) = − L
R 2 s + 1
Rs 2 + L 2
R 2 (R +Ls)
InreadinessfortakinginverseLaplacetransformswewritethefinaltermasfollows:
L 2
R 2 (R +Ls) = L 2
R 2 L (R/L +s) = L
R 2 (R/L +s)
Hence
I(s) = − L
R 2 s + 1
Rs + L
2 R 2 (R/L +s)
Taking the inverse Laplace transformyields
i(t) = − L R 2 + t R + L R 2 e−(R/L)t t 0
This may berearranged as
i(t) = t R + L R 2 (e−(R/L)t −1)
t 0
Example21.23 Solve
y ′′ −y=−t 2 y(0) =2, y ′ (0) =0
using the Laplace transform.
Solution Let L(y) =Y(s). Thenusingthe resultstated inSection21.5 wehave
L(y ′′ ) =s 2 Y(s) −sy(0) −y ′ (0) =s 2 Y(s) −2s