Higher Engineering Mathematics, Sixth Edition
Chapter 61 Introduction to Laplace transforms 61.1 Introduction The solution of most electrical circuit problems can be reduced ...
Introduction to Laplace transforms 583 (a) f(t)= 1. From equation (1), L{ 1 }= ∫∞ 0 e−st( 1 )dt= [ e−st −s ]∞ 0 =− 1 s [e−s(∞)−e ...
584 Higher Engineering Mathematics Table 61.1Elementary standard Laplace transforms Function Laplace transforms f(t) L{f(t)}= ∫∞ ...
Introduction to Laplace transforms 585 = 1 s^2 +a^2 [( 0 )− 1 ( 0 −a)] = a s^2 +a^2 (provideds> 0 ) (b) From equation (1), L{ ...
586 Higher Engineering Mathematics (a) 2t−3(b)5t^2 + 4 t− 3 [ (a) 2 s^2 − 3 s (b) 10 s^3 + 4 s^2 − 3 s ] (a) t^3 24 − 3 t+2( ...
Chapter 62 Properties of Laplace transforms 62.1 The Laplace transform ofeatf(t) FromChapter61,thedefinitionoftheLaplacetransfor ...
588 Higher Engineering Mathematics Problem 1. Determine (a)L{ 2 t^4 e^3 t} (b)L{4e^3 tcos5t}. (a) From (i) of Table 62.1, L{ 2 t ...
Properties of Laplace transforms 589 = 3 2 s+ 1 − 6 s+ 3 4 s^2 + 4 s+ 17 = 3 ( 4 s^2 + 4 s+ 17 )−( 6 s+ 3 )( 2 s+ 1 ) ( 2 s+ 1 ) ...
590 Higher Engineering Mathematics (b) Second derivative Let the second derivative off(t)bef′′(t), then from equation (1), L{f′′ ...
Properties of Laplace transforms 591 Use the Laplace transform of the first deriva- tive to derive the transforms: (a)L{eat}= ...
592 Higher Engineering Mathematics i.e. 3e−∞=( 0 ) ( 3 0 + 4 ) i.e. 0 = 0 , which illustrates the theorem. Problem 9. Verify the ...
Chapter 63 Inverse Laplace transforms 63.1 Definition of the inverse Laplace transform If the Laplace transform of a function f( ...
594 Higher Engineering Mathematics HenceL−^1 { 1 s^2 + 9 } =L−^1 { 1 s^2 + 32 } = 1 3 L−^1 { 3 s^2 + 32 } = 1 3 sin3t (b)L−^1 { ...
Inverse Laplace transforms 595 (a)L−^1 { 3 s^2 − 4 s+ 13 } =L−^1 { 3 (s− 2 )^2 + 32 } =e^2 tsin3t, from (xii) of Table 63.1 (b)L ...
596 Higher Engineering Mathematics (a) s+ 1 s^2 + 2 s+ 10 (b) 3 s^2 + 6 s+ 13 [ (a)e−tcos3t (b) 3 2 e−^3 tsin2t ] (a) 2 (s− ...
Inverse Laplace transforms 597 Problem 9. Determine L−^1 { 5 s^2 + 8 s− 1 (s+ 3 )(s^2 + 1 ) } 5 s^2 + 8 s− 1 (s+ 3 )(s^2 + 1 ) ≡ ...
598 Higher Engineering Mathematics 7. 26 −s^2 s(s^2 + 4 s+ 13 ) [ 2 −3e−^2 tcos3t− 2 3 e−^2 tsin3t ] 63.4 Poles and zeros It was ...
Inverse Laplace transforms 599 Problem 12. Determine the poles and zeros for the function:F(s)= (s+ 3 )(s− 2 ) (s+ 4 )(s^2 + 2 s ...
Chapter 64 The solution of differential equations using Laplace transforms 64.1 Introduction An alternative method of solving di ...
The solution of differential equations using Laplace transforms 601 (ii) y( 0 )=4andy′( 0 )= 9 Thus 2[s^2 L{y}− 4 s−9]+5[sL{y}−4 ...
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