Engineering Mathematics 1
Page 3 of 11
2. (a) Find the general solution of
0 1
y =
y.
−4 0
0
Write your answer in terms of real-valued functions.
(b) Write the third-order ODE
y 000 + y 00 + y = 0
as a system of first-order ODEs of the form y 0 = Ay. Make sure you clearly define
the components of A and y. Do not solve the resulting system.
[15 marks]
3. Consider the second-order ordinary differential equation
xy 00 − 2(x − 1)y 0 + (x − 2)y = 0,
x > 0.
(2)
(a) Since x = 0 is a regular singular point of equation (2), the equation has at least one
solution of the form
∞
X
y=
am xm+r .
(3)
m=0
0
00
Write down expressions for y and y .
(b) Substituting equation (3) and its derivatives into equation (2), rearranging some terms
and shifting some indices yields the intermediate result
∞
X
m=0
am (m + r)(m + r + 1)x
m+r−1
−2
∞
X
am (m + r + 1)x
m=0
m+r
+
∞
X
am−1 xm+r = 0. (4)
m=1
Do not derive this result. Use this result to find the indicial equation and solve it
for r. Hint: You will need to shift indices in the first summation.
(c) For r = 0, find an expression for a1 in terms of a0 and write down the recurrence
relation for the remaining coefficients.
(d) Calculate the first three terms of the solution for r = 0.
[15 marks]
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