Engineering Mathematics III
Summative Assignment
Question 1
a) Let 𝐶 be defined by,
𝒓(𝑡) = {
(cos 𝑡 , sin 𝑡 , 0), 0 ≤ 𝑡 ≤
1 3√3
(− 2 ,
2
𝑡
(1 − 𝜋) , 0) ,
2𝜋
3
2𝜋
3
≤𝑡≤𝜋
Determine whether 𝒓(𝑡) is a smooth or discontinuous curve.
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b) Find the arc-length of the curve defined by 𝑥 = 2𝑡, 𝑦 = 𝑡 2 and 𝑧 = ln 𝑡 , 𝑡 > 0,
between the points (2,1,0) and (4,4, ln 2).
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Question 2
a) Given 𝑤 = 𝑓(𝑥, 𝑦); 𝑥 = 𝑟𝑐𝑜𝑠𝜃, 𝑦 = 𝑟𝑠𝑖𝑛𝜃; show that
𝜕𝑤 2
1
𝜕𝑤 2
𝜕𝑤 2
𝜕𝑤 2
( 𝜕𝑟 ) + 𝑟 2 ( 𝜕𝜃 ) = ( 𝜕𝑥 ) + ( 𝜕𝑦 ) .
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b) The voltage 𝑉 in a circuit that satisfies the law 𝑉 = 𝐼𝑅 is slowly dropping as the
battery wears out. At the same time, the resistance 𝑅 is increasing as the resistor
heats up. Use the equation
𝑑𝑉 𝜕𝑉 𝑑𝐼 𝜕𝑉 𝑑𝑅
=
+
𝑑𝑡
𝜕𝐼 𝑑𝑡 𝜕𝑅 𝑑𝑡
to find how the current is changing at the instant when
𝑑𝑅
𝑅 = 600 𝑜ℎ𝑚𝑠, 𝐼 = 0.04𝑎𝑚𝑝, 𝑑𝑡 = 0.5 𝑜ℎ𝑚 𝑝𝑒𝑟 𝑠𝑒𝑐,
and
𝑑𝑉
𝑑𝑡
= −0.01𝑣𝑜𝑙𝑡 𝑝𝑒𝑟 𝑠𝑒𝑐.
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Question 3
Consider the ordinary differential equation (ODE)
2𝑥𝑦 ′′ + (1 + 𝑥)𝑦 ′ + 3𝑦 = 0,
in the neighbourhood of the origin.
a) Show that 𝑥 = 0 is a regular singular point of the ODE.
b) By seeking an appropriate solution to the ODE, show that
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i) the roots to the indicial equation of the ODE are 0 and 1/2.
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ii) the recurrence formula used to determine the power series coefficients, 𝑐𝑛 ,
when one of the indicial equation roots is 1/2 is given by
𝑐1 = −
(2𝑠+7)𝑐𝑠
7𝑐0
6
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𝑐𝑠+1 = − (2𝑠+3)(𝑠+1) , 𝑠 = 1,2, … . .
iii) For the indicial equation root 1/2, show that one of the solutions to the ODE is
given by
𝑦 = 𝑐0 √𝑥 [1 −
7𝑥
6
+
21𝑥 2
40
−
77𝑥 3
560
+⋯]
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Question 4
a) Using the Cauchy’s integral formula, evaluate
𝐼 = ∮𝐶
2𝑧
(1−3𝑧)(𝑧+2)
𝑑𝑧, where 𝐶 is a circle |𝑧| = 3.
10]
𝑧 2 +1
b) i) Find the residues of the function 𝑓(𝑧) = 𝑧(𝑧−6) at its poles in the finite
complex plane.
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ii)Using the residues in (4(b)(i), evaluate the integral ∮𝐶
𝑧 2 +1
𝑧(𝑧−6)
𝑑𝑧
where C is a contour that includes all the singularities of 𝑓(𝑧) and is
in the positive sense.
2
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