Higher Engineering Mathematics

622 STATISTICS AND PROBABILITY Table 63.5 Critical values for the Mann-Whitney test α 1 =5% 212 %1%^12 % α 1 =5% 212 %1%^12 % n ...

CHI-SQUARE AND DISTRIBUTION-FREE TESTS 623 J Table 63.5 (Continued) α 1 =5% 212 %1%^12 % α 1 =5% 212 %1%^12 % n 1 n 2 α 2 =10% 5 ...

624 STATISTICS AND PROBABILITY 41875 1410 12 5 20 21 25 28 34 0 10 20 30 1196 16 18 SAMPLE P SAMPLE Q Figure 63.1 Now try the fo ...

Assign-17-H8152.tex 23/6/2006 15: 16 Page 625 J Statistics and probability Assignment 17 This assignment covers the material con ...

Assign-17-H8152.tex 23/6/2006 15: 16 Page 626 626 STATISTICS AND PROBABILITY determine if the distribution gives a ‘too good’ fi ...

Laplace transforms K 64 Introduction to Laplace transforms 64.1 Introduction The solution of most electrical circuit problems ca ...

628 LAPLACE TRANSFORMS From equation (1), L{eat}= ∫∞ 0 e−st(eat)dt= ∫∞ 0 e−(s−a)tdt, from the laws of indices, = [ e−(s−a)t −(s− ...

INTRODUCTION TO LAPLACE TRANSFORMS 629 K 64.5 Worked problems on standard Laplace transforms Problem 1. Using a standard list of ...

630 LAPLACE TRANSFORMS = [ (0− 0 −0)− ( 0 − 0 − 2 s^3 )] = 2 s^3 (provideds>0) (c) From equation (1), L{coshat}=L { 1 2 (eat+ ...

INTRODUCTION TO LAPLACE TRANSFORMS 631 K (a) 5e^3 t(b) 2e−^2 t [ (a) 5 s− 3 (b) 2 s+ 2 ] (a) 4 sin 3t(b) 3 cos 2t [ (a) 12 s ...

Laplace transforms 65 Properties of Laplace transforms 65.1 The Laplace transform of eatf(t) From Chapter 64, the definition of ...

PROPERTIES OF LAPLACE TRANSFORMS 633 K = 4(s−3) s^2 − 6 s+ 9 + 25 = 4(s−3) s^2 − 6 s+ 34 Problem 2. Determine (a)L{e−^2 tsin 3t} ...

634 LAPLACE TRANSFORMS Now try the following exercise. Exercise 232 Further problems on Laplace transforms of the form eatf(t) D ...

PROPERTIES OF LAPLACE TRANSFORMS 635 K assuming e−stf′(t)→0ast→∞, andf′(0) is the value off′(t)att=0. Hence {f′′(t)}=−f′(0)+s[s( ...

636 LAPLACE TRANSFORMS Use the Laplace transform of the second derivative to derive the transforms: (a)L{sinhat}= a s^2 −a^2 ( ...

PROPERTIES OF LAPLACE TRANSFORMS 637 K Problem 9. Verify the final value theorem for the function (2+3e−^2 tsin 4t) cm, which re ...

Laplace transforms 66 Inverse Laplace transforms 66.1 Definition of the inverse Laplace transform If the Laplace transform of a ...

INVERSE LAPLACE TRANSFORMS 639 K Problem 2. Find the following inverse Laplace transforms: (a)L−^1 { 6 s^3 } (b) L−^1 { 3 s^4 } ...

640 LAPLACE TRANSFORMS (a)L−^1 { 5 s^2 + 2 s− 3 } =L−^1 { 5 (s+1)^2 − 22 } =L−^1 ⎧ ⎪⎨ ⎪⎩ 5 2 (2) (s+1)^2 − 22 ⎫ ⎪⎬ ⎪⎭ = 5 2 e−ts ...

INVERSE LAPLACE TRANSFORMS 641 K simpler fractions which may be inverted on sight. For example, the function, F(s)= 2 s− 3 s(s−3 ...