Understanding Engineering Mathematics

17.4 Properties of the Laplace transform Having seen how to obtain elementary Laplace transforms we now turn to the general prop ...

Note how differentiation with respect totin ‘t-space’ becomes multiplication bysin ‘s-space’. Also, note the appearance of the i ...

Sincey( 0 )=0 this becomes (s+ 1 )y(s) ̃ = 1 s Solving fory(s) ̃ now gives y(s) ̃ = 1 s(s+ 1 ) Now by separation of variables (4 ...

method of solution – we do not have to find the general solution first and then apply the initial conditions afterwards. Exercis ...

(i) 2 s^2 − 3 s+ 1 2 s^3 (ii) 1 s+ 1 (iii) 1 s− 3 (iv) 4 2 s− 1 (v) 5 s^2 + 4 (vi) 3 s s^2 + 9 (vii) s+ 1 s^2 − 5 s+ 6 (viii) 1 ...

Take the Laplace transform of the equation, inserting the initial values. Solve for the transform of the solution, obtaining y ...

Answers (i) 10 3 e−^3 t+ 2 3 (ii) et− 1 −t 10e−t− 20 e−^2 t+ 10 e−^3 t 17.7 Linear systems and the principle of superpositio ...

t = −aw 0 t A 2 p w Figure 17.6Sinusoidal functionAsin(ωt+α). Exercise on 17.7 Show that (i) the sum of two sinusoids with the s ...

Ifm,nare any integers, then: ∫π −π sinmtsinntdt=0ifm=n ∫π −π sin^2 ntdt=π ∫π −π cosmtcosntdt=0ifm=n ∫π −π cos^2 ntdt=π ∫π −π s ...

increasing number of times in this period. The above series is therefore adding together a (possibly infinite) number of differe ...

There are a number of points that are worth noting, although we do not want to be too picky at this stage, so treat these as ref ...

Answer − 3 p − 2 p −p 0 p 2 p 3 p t Even; π 2 ;− 4 π ;0ifnis even, and− 4 π( 2 r+ 1 )^2 ifn= 2 r+1 is odd 17.10 The Fourier coef ...

In general, we have an= 1 π ∫π −π f(t)cosntdtn= 0 , 1 , 2 ,... bn= 1 π ∫π −π f(t)sinntdtn= 1 , 2 ,... Problem 17.10 Verify the s ...

So we get πbm= ∫π −π f(t)sinmtdt = ∫ 0 −π (−A)sinmtdt+ ∫π 0 Asinmtdt =A [ cosmt m ] 0 −π −A [ cosmt m ]π 0 = A m ( 1 −(− 1 )m)− ...

17.11 Reinforcement 1.Find the Laplace transforms of (i) 2+t+ 3 t^2 (ii) 2 sin 3t+e−^2 t (iii) etcos( 2 t) (iv) sin^2 t 2.Write ...

5.State which of the following functions oftare periodic and give the period when they are. (i) tan 2t (ii) cos ( 3 πt 2 ) (iii) ...

equations and studying the properties of the solutions. This question is a significant project in which you are asked to repeat ...

i.e. x ̃o(s)= G(s) 1 +F(s)G(s) x ̃i(s) x∼o x∼f x∼i x∼∼i − xf x o ∼ Feedback Control F(s) G(s) Figure 17.8 By such means we can d ...

periodic motion – the energy/power in any periodic phenomenon is proportional to the sum of the squares of the amplitudes of the ...

(i) π− 8 π ∑∞ n= 1 cos( 2 n− 1 )t ( 2 n− 1 )^2 (ii) 2 ∑∞ n= 1 (− 1 )n n sinnt (iii) π^2 3 + 4 ∑∞ n= 1 (− 1 )n n^2 cosnt (iv) π ...