CITY UNIVERSITY LONDON

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1 No: CITY UNIVERSITY LONDON BEng (Hons)/MEng (Hons) Degree in Civil Engineering BEng (Hons)/MEng (Hons) Degree in Civil Engineering with Surveying BEng (Hons)/MEng (Hons) Degree in Civil Engineering with Architecture PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (CIVIL) EX22 Date: August 24 Time: 2 hours Attempt Three out of FIVE questions

2 Question 1 (a) Using Laplace transforms (or otherwise) solve the following differential equation: subject to the initial conditions: d 2 y(t) dy(t) + y(t) = e t y() = 1 and dy() = 1 Check your solution by substituting into the differential equation and by verifying that the initial conditions are satisfied. A table of Laplace transforms is provided at the end of the paper. [1 marks] (b) Show from first principles (i.e. without reference to the table of Laplace transforms) that: L(cos ωt) = s s 2 + ω 2 Indicate clearly the region of convergence of the transform, i.e. the range of values of s for which your result is valid. Hint: Write cos ωt = 1 2 (ejωt + e jωt ) and apply the definition of the transform. (c) Consider the function f(t) = t sin ωt (t ), f(t) = (t < ). Show by direct differentiation that f (t) = 2ω cos ωt ω 2 f(t) By using the properties of Laplace transforms of derivatives, show that L(t sin ωt) = 2ωs (s 2 + ω 2 ) 2 In your derivation you will need to use the Laplace transform of the function cos ωt you obtained in part (b). Question 2 (a) On the linear space R[, 2π] (real valued-functions defined on the interval [, 2π]) define the inner product of two functions f and g as: f, g = 2π f(x)g(x)dx A set S of functions in R[, 2π] is said to be orthonormal if: (i) f, g = for any two functions f and g in S such that f g, and (ii) f, f = 1 for every function in f in S. Prove that the set of functions: S = { 1 2π, cos x π, } sin x π is orthonormal. [8 marks] 2 of 5

3 (b) Consider the periodic function f(t) with period T = 2π, defined as f(t) = t 2 in the interval π < t π. The Fourier series expansion of f(t) is of the form: f(t) = a 2 + a n cos nt + where the a n s and b n s are unspecified coefficients. b n sin nt Show that f(t) is an even function and, as a result, b n = for all n > in the Fourier series expansion of f(t). Calculate the a n s (n ) in closed form and hence show that: f(t) = π ( 1) n cos(nt) n 2 Hint: You need to integrate by parts (twice). Show that: ( 1) n+1 Hint: Set t = in your Fourier series expansion. n 2 = = π2 12 [8 marks] Question 3 (a) Define the following terms of linear-algebra: A subspace of a vector space V. Direct sum of two subspaces. Linear independence of a list of vectors. Linear span of a list of vectors. Basis of a vector space V. Range and Kernel of a linear transformation. Give simple examples to illustrate your definitions. (b) Show that: (i) The intersection of two subspaces of a vector space V is a subspace of V. (ii) A list of vectors containing two identical vectors is linearly dependent. (c) Let S 2 2 denote the set of 2 2 symmetric matrices with real entries (A matrix A is called symmetric if A = A T where A T is the transpose of A). For a fixed 2 2 symmetric matrix A define the transformation Π A : S 2 2 S 2 2 which maps 2 2 symmetric matrices X to 3 of 5

4 2 2 symmetric matrices Y according to the rule Y = AXA T. Show that Π A is a linear transformation. If: ( A = ) find the Range and Kernel of Π A, and hence verify the rank-nullity theorem. [8 marks] Question 4 (a) We wish to perform the following three elementary operations on the rows of a 3 3 matrix A: Multiply the first row by 2. Interchange the first and third rows. Add twice the second row to the third row. Write down the three elementary matrices by which A must be pre-multiplied to perform each operation. If the three operations must be performed in sequence, find the overall transformation matrix and its inverse. (b) Consider the linear system of equations Ax = b, where A R m n and x is the vector of unknowns. Show that the system is consistent (i.e. has at least one solution) if and only if Rank[A] = Rank[A b]. Show also that a sufficient and necessary condition for this system of equations to have exactly one solution is that Rank[A] = Rank[A b] = n. [8 marks] (c) Find all solutions of the system of equations: x y = z 3 Explain clearly why your result is consistent with the conditions in part (b) above. Question 5 (a) Define the convolution f(t) g(t) of two functions f(t) and g(t) (such that f(t) = g(t) = for t < ). What is the Laplace transform of f(t) g(t)? (b) Consider the initial value problem: dy(t) + ay(t) = u(t), y() = y where a is a real parameter and u(t) is a known function. By taking Laplace transforms (or otherwise) show that the solution to this problem is y(t) = e at y + e at u(t). 4 of 5

5 (c) By using the convolution property of the Laplace transform find all solutions of the following integral equation: te at = t y(t τ)y(τ)dτ where y(t) = for t <. Check your solution by substitution into the right-hand-side of this equation and working out the corresponding integral. [7 marks] (d) The matrix inversion lemma states that for four matrices A, B, C and D of compatible dimensions the following identity holds, provided the indicated inverses exist: (A BD 1 C) 1 = A 1 + A 1 B(D CA 1 B) 1 CA 1 By multiplying the matrix in the right-hand-side of the above equation by A BD 1 C (either from the left or the right) show that this identity is valid. What are the computational advantages of using this identity when A = I n, B is a column vector and C is a row vector (and hence D is scalar)? Table of Laplace Transforms f(t) F (s) f(t) F (s) s δ(t) 1 cos ωt s 2 +ω 2 ω 1 1/s sin ωt s 2 +ω 2 t 1/s 2 s cosh at s 2 a 2 t 2 2/s 3 a sinh at s 2 a 2 t n n! e at s a cos ωt s n+1 (s a) 2 +ω 2 e at 1 e at ω sin ωt s a (s a) 2 +ω 2 te at 1 t n 1 e at (n 1)! (s+a) 2 (s+a) n External Examiners: Professor M R Barnes, Mr S Barnes Internal Examiner: Dr G. Halikias 5 of 5