9231/22 Worked Solutions

Question 1 - Hyperbolic functions [4]

(a) Using the identity cosh^2 x - sinh^2 x = 1:

cosh^2 x = 1 + (3/4)^2 = 1 + 9/16 = 25/16
cosh x = 5/4 (positive root, as cosh x >= 1)

(b) tanh x = (3/4)/(5/4) = 3/5

(c) Using sinh 2x = 2 sinh x cosh x:

sinh 2x = 2(3/4)(5/4) = 15/8

Question 2 - Eigenvalues and eigenvectors [10]

(a) Find determinant of (A - lambda I):

det = (2-lambda)(3-lambda)^2 - 1(3-lambda) + 1[-(3-lambda)]
= (2-lambda)(3-lambda)^2 - 2(3-lambda)
= (3-lambda)[(2-lambda)(3-lambda) - 2]
= (3-lambda)(6 - 5lambda + lambda^2 - 2)
= (3-lambda)(lambda^2 - 5lambda + 4)
= (3-lambda)(lambda-1)(lambda-4) = 0

Hence eigenvalues: lambda = 1, 3, 4.

(b) For each eigenvalue, solve (A - lambda I)v = 0:

lambda = 1:

v1 + 2v2 = 0, v1 + 2v3 = 0 => v2 = v3 = -v1/2
Eigenvector: (2, -1, -1)^T

lambda = 3:

v1 = 0, v2 + v3 = 0 => v3 = -v2
Eigenvector: (0, 1, -1)^T

lambda = 4:

v1 = v2, v1 = v3 => v1 = v2 = v3
Eigenvector: (1, 1, 1)^T

(c) Q = [[2,0,1],[-1,1,1],[-1,-1,1]], D = diag(1,3,4)

Question 3 - Differential equations [8]

Auxiliary equation: m^2 + 2m + 5 = 0

m = -1 +/- 2i
y_c = e^{-x}(A cos 2x + B sin 2x)

Particular integral: Try y_p = p cos x + q sin x

Substitute: cos terms: -p+2q+5p = 4p+2q = 0
sin terms: -q-2p+5q = -2p+4q = 10
Solving: p = -1, q = 2, so y_p = -cos x + 2 sin x

General solution: y = e^{-x}(A cos 2x + B sin 2x) - cos x + 2 sin x

Initial conditions:

x=0, y=0: A - 1 = 0 => A = 1
x=0, y'=2: -A + 2B + 2 = 2 => B = 1/2

Particular solution: y = e^{-x}(cos 2x + 0.5 sin 2x) - cos x + 2 sin x

Question 4 - Arc length and surface area [9]

(a) y = cosh x, dy/dx = sinh x

L = integral sqrt(1 + sinh^2 x) dx = integral cosh x dx
= [sinh x] = sinh 1 = (e - e^{-1})/2

(b) Surface area of revolution:

S = 2pi integral y sqrt(1 + (y')^2) dx = 2pi integral cosh^2 x dx
= 2pi integral (1 + cosh 2x)/2 dx
= pi [x + (1/2) sinh 2x]
= pi(1 + sinh 2 / 2)

Question 5 - Reduction formula [10]

(a) I_n = integral x^n cos x dx from 0 to pi/2

IBP: u=x^n, dv=cos x dx
I_n = [x^n sin x] - n integral x^{n-1} sin x dx
= (pi/2)^n - n integral x^{n-1} sin x dx
IBP again: u=x^{n-1}, dv=sin x dx
integral x^{n-1} sin x dx = [-x^{n-1} cos x] + (n-1) integral x^{n-2} cos x dx
= (n-1)I_{n-2}
Hence I_n = (pi/2)^n - n(n-1)I_{n-2}

(b)

I_0 = integral cos x dx = [sin x] = 1
I_2 = (pi/2)^2 - 2(1)I_0 = pi^2/4 - 2
I_4 = (pi/2)^4 - 4(3)I_2 = pi^4/16 - 12(pi^2/4 - 2)
= pi^4/16 - 3pi^2 + 24

Question 6 - Complex numbers (De Moivre) [10]

(a) Express sin^5 theta using de Moivre.

Let z = cos theta + i sin theta. Then z - z^{-1} = 2i sin theta
(2i sin theta)^5 = (z - z^{-1})^5
= z^5 - 5z^3 + 10z - 10z^{-1} + 5z^{-3} - z^{-5}
32i sin^5 theta = (z^5 - z^{-5}) - 5(z^3 - z^{-3}) + 10(z - z^{-1})
= 2i sin 5theta - 5(2i sin 3theta) + 10(2i sin theta)

Result: sin^5 theta = (1/16)(sin 5theta - 5 sin 3theta + 10 sin theta)

(b) Evaluate the integral:

integral sin^5 theta dtheta = (1/16) integral (sin 5theta - 5 sin 3theta + 10 sin theta) dtheta
= (1/16)[-cos 5theta/5 + 5 cos 3theta/3 - 10 cos theta]
At theta=pi/2: all cosine terms = 0
At theta=0: -1/5 + 5/3 - 10 = -128/15
Result: (1/16)(0 + 128/15) = 8/15

Question 7 - Cayley-Hamilton theorem [11]

(a) Characteristic equation:

det(B - lambda I) = (1-lambda)[(1-lambda)^2 - 4] - 2[2(1-lambda)]
= (1-lambda)(lambda^2 - 2lambda - 3) - 4(1-lambda)
= (1-lambda)(lambda^2 - 2lambda - 7)
= -lambda^3 + 3lambda^2 + 5lambda - 7

Characteristic equation: lambda^3 - 3lambda^2 - 5lambda + 7 = 0

(b) Verify CH: B^3 - 3B^2 - 5B + 7I = 0

B^2 = [[5,4,4],[4,9,4],[4,4,5]]
B^3 = [[13,22,12],[22,25,22],[12,22,13]]
B^3 - 3B^2 - 5B + 7I = 0 (verified by direct computation)

(c) Find B^{-1}:

B^2 - 3B - 5I + 7B^{-1} = 0
7B^{-1} = -B^2 + 3B + 5I

B^{-1} = (1/7)[[3,2,-4],[2,-1,2],[-4,2,3]]

(d) Express B^4:

From CH: B^3 = 3B^2 + 5B - 7I
B^4 = B * B^3 = 3B^3 + 5B^2 - 7B
= 3(3B^2 + 5B - 7I) + 5B^2 - 7B
= 9B^2 + 15B - 21I + 5B^2 - 7B

B^4 = 14B^2 + 8B - 21I (so p=14, q=8, r=-21)

Question 8 - Inverse hyperbolic and Maclaurin [13]

(a) sinh^{-1} x = ln(x + sqrt(x^2 + 1))

(b) f'(x) = 1/sqrt(x^2 + 1) (standard derivative)

(c) Maclaurin series up to x^3:

f(0) = 0
f'(x) = (1+x^2)^(-1/2), f'(0) = 1
f''(x) = -x(1+x^2)^(-3/2), f''(0) = 0
f'''(x) = -(1+x^2)^(-3/2) + 3x^2(1+x^2)^(-5/2), f'''(0) = -1

sinh^{-1} x = x - x^3/6 + ...

(d) Estimate sinh^{-1}(0.4):

sinh^{-1}(0.4) ~= 0.4 - 0.4^3/6 = 0.4 - 0.064/6
= 0.4 - 0.010667 = 0.389333...

~= 0.3893 (4 d.p.)

Actual value: 0.3900. Error approx 0.0007.