Understanding Engineering Mathematics

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) cos(

√

t)

(iv) sint+cos 2t (v) sin|t| (vi) |cost|

(vii) sin

(

2 π

L

t

)

(viii) cos( 4 ωt) (ix) 4 cos 2t+3sin4t

(x) t^2 cost.

6.What is the period of

f(t)=

1

2

a 0 +

∑∞

n= 1

(ancosnωt+bnsinnωt)?

7.Obtain the Fourier series for the following functions defined on−π<t<π:

(i) 2|t|−π<t<π (ii) t −π<t≤π

(iii) t^2 −π<t<π (iv) f(t)= 0 −π<t< 0

=t 0 <t<π

(v) f(t)=−t^2 −π<t<0(vi)f(t)= 0 −π<t< 0

=t^20 <t<π = 10 <t<π

(vii) f(t)=













1

2

−π<t<−

π

2

1 −

π

2

<t<

π

2

0

π

2

<t<π

17.12 Applications

Transform methods form a vast area of engineering mathematics, and this chapter only
scratches the surface. In these applications we just flag up some of the key fundamental
ideas which may come up in your engineering subjects.

1.One of the most important applications of Laplace transforms in engineering is in
the solution of initial value problems of the sort discussed in Chapter 15. Particularly
important are the sorts of engineering models described by inhomogeneous second order
differential equations of the type covered in Chapter 15, Applications, question 6:

mx ̈+βx ̇+αx=f(t)

In Chapter 15 we solved this type of equation by the auxiliary equation and the method

of undetermined coefficients. In the case when we have initial conditions, specifying

xandx ̇ att=0, the Laplace transform provides a powerful tool for solving such