Higher Engineering Mathematics

642 LAPLACE TRANSFORMS HenceL−^1 { 5 s^2 + 8 s− 1 (s+3)(s^2 +1) } ≡L−^1 { 2 s+ 3 + 3 s− 1 s^2 + 1 } ≡L−^1 { 2 s+ 3 } +L−^1 { 3 s ...

INVERSE LAPLACE TRANSFORMS 643 K Poles: The valuesa,b,c, ... that makes the denomi- nator zero, and hencef(s) infinite, are call ...

644 LAPLACE TRANSFORMS − 4 − 3 − 20 − 1123 σ jω j −j Figure 66.2 Now try the following exercise. Exercise 237 Further problems o ...

K Laplace transforms 67 The solution of differential equations using Laplace transforms 67.1 Introduction An alternative method ...

646 LAPLACE TRANSFORMS Hencey=L−^1 { 8 s+ 38 2 s^2 + 5 s− 3 } =L−^1 { 12 2 s− 1 − 2 s+ 3 } =L−^1 { 12 2 ( s−^12 ) } −L−^1 { 2 s+ ...

THE SOLUTION OF DIFFERENTIAL EQUATIONS USING LAPLACE TRANSFORMS 647 K Equating s^2 terms gives: 0=A+C, from which,A=−1, sinceC=1 ...

648 LAPLACE TRANSFORMS (ii)i(0)=0, henceRL{i}+LsL{i}= E s (iii) Rearranging gives: (R+Ls)L{i}= E s i.e. L{i}= E s(R+Ls) (iv)i=L− ...

THE SOLUTION OF DIFFERENTIAL EQUATIONS USING LAPLACE TRANSFORMS 649 K d^2 y dx^2 + dy dx − 2 y=3 cos 3x−11 sin 3x, giveny(0) ...

Laplace transforms 68 The solution of simultaneous differential equations using Laplace transforms 68.1 Introduction It is somet ...

THE SOLUTION OF SIMULTANEOUS DIFFERENTIAL EQUATIONS USING LAPLACE TRANSFORMS 651 K and equation (2′) becomes: sL{x}−L{y}=− 4 s− ...

652 LAPLACE TRANSFORMS Equation (1) becomes: 3[sL{x}−x(0)]−5[sL{y}−y(0)] + 2 L{x}= 6 s from equation (3), page 634, and Table 64 ...

THE SOLUTION OF SIMULTANEOUS DIFFERENTIAL EQUATIONS USING LAPLACE TRANSFORMS 653 K Returning to equations (A) to determineL{x}an ...

654 LAPLACE TRANSFORMS Adding equations (5) and (6) gives: [(s^2 +1)(s^2 −1)+1]L{x}=(s^2 +1)2s−s i.e. s^4 L{x}= 2 s^3 +s=s(2s^2 ...

Assign-18-H8152.tex 23/6/2006 15: 17 Page 655 K Laplace transforms Assignment 18 This assignment covers the material contained i ...

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Fourier series L 69 Fourier series for periodic functions of period 2π 69.1 Introduction Fourier seriesprovides a method of anal ...

658 FOURIER SERIES (ii)a 0 ,anandbnare called theFourier coefficients of the series and if these can be determined, the series o ...

FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD 2π 659 L Hencea 1 ,a 2 ,a 3 ,...are all zero (since sin 0= sin (−nπ)=sinnπ=0), a ...

660 FOURIER SERIES −π −π/ 2 0 π/ 2 π −π − 4 π 4 f(x) P 1 f(x) x −π −π/ 2 0 π/ 2 π x P 2 P 1 f(x) f(x) π −π 4 /3 sin 3x f(x) π −π ...

FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD 2π 661 L = 5 2 π {[ −cos [ 2 π ( 1 2 +n )] ( 1 2 +n ) − cos [ 2 π ( 1 2 −n )] ( ...