Data provided: Formua Sheet MAS250 SCHOOL OF MATHEMATICS AND STATISTICS Mathematics II (Materias Autumn Semester 204 5 2 hours Marks wi be awarded for answers to a questions in Section A, and for your best THREE answers to questions in Section B. Section A carries 40 marks, and the marks awarded to each question or section of question are shown in itaics. Section A A Find the genera soution of the equation x 2 dy e /x y = x (7 marks A2 Find the particuar soution of the equation d 2 y + 4dy + 3y = 2 2e 3x which satises dy = 0 and y = when x = 0. (3 marks A3 The pressure (p, voume (V and temperature (T of a container fu of gas are reated by the idea gas aw pv = nrt, where n measures the number of gas partices and R is the universa gas constant. The pressure is initiay 0 5 Nm 2. Using the chain rue, estimate the pressure if the voume decreases by 4% and the temperature increases by 3%. (7 marks MAS250 Turn Over
A4 Two quantities x and y have means 2.36 and 73.4 respectivey, standard deviations 3.8 and 2.9 respectivey, and covariance -6.87. (a Cacuate the correation coecient between x and y, correct to 3 signicant gures. (2 marks (b It is assumed that x and y satisfy the inear reationship y = a + b(x x, ( where x is the mean of x. Cacuate the east squares estimates of a and b, correct to 3 signicant gures. State, giving a reason, whether you expect ( to give a good mode. (4 marks A5 Find a vector norma to the surface ϕ = 5 at the point A with coordinates (, 2, 3, where ϕ = x 2 cos πz+xyz z sin πy. (5 marks Find aso the directiona derivative of ϕ at A, in the direction d = (2, 0,. (2 marks Section B B (a For x > 0, nd the particuar soution of the equation x dy = y + x3 y 2 which satises y = 3 when x =. ( marks (b Find the genera soution of the equation 4 d2 y +4dy +5y = x. 2 (9 marks B2 (a Let R = (x 2 + y 2 /2, and dene a scaar ed ϕ by Find ϕ, and show that ϕ = n R. 2 ϕ = 0. (2 marks (b A vector ed u is given by u = (x + y sin z, y + e z, xz cosh y. Verify that and nd u. ( u = 0, (8 marks MAS250 2 Continued
B3 A function f(x = x 2 is dened on the interva 0 x. (a Show that f(x can be represented by the Fourier series 3 + 4 π 2 ( n cos(nπx. n= n 2 [You may use the fact that cos(nπ = ( n.] (5 marks (b Use the resut of part (a to nd ( n n= n 2. (5 marks B4 The function ϕ(x, y satises Lapace's equation in two dimensions, i.e. 2 ϕ x + 2 ϕ 2 y = 0, 2 in the square region 0 < x <, 0 < y <. The boundary conditions are ϕ(0, y = 0 ϕ(x, 0 = 0 ϕ(x, = 0 ϕ(, y =. If ϕ(x, y = X(xY (y for some functions X and Y, show that X X = Y Y = α, where α must be a constant. Find the vaues of X(0, Y (0 and Y (. (3 marks (3 marks Given that α > 0, deduce that ϕ(x, y = A n sinh nπx sin nπy n= for some constants A n. Show that 0 n even A n = 4 n odd. nπ sinh nπ (8 marks (6 marks End of Question Paper MAS250 3 Turn Over
FORMULA SHEET Trigonometry + tan 2 θ = sec 2 θ + cot 2 θ = cosec 2 θ cos(a + B = cos A cos B sin A sin B cos(a B = cos A cos B + sin A sin B sin(a + B = sin A cos B + cos A sin B sin(a B = sin A cos B cos A sin B tan(a + B = tan(a B = sin 2θ = 2 sin θ cos θ tan A + tan B tan A tan B tan A tan B + tan A tan B cos 2θ = 2 cos 2 θ = 2 sin 2 θ a cos θ + b sin θ = R cos(θ α, where R = (a 2 + b 2, cos α = a/r and sin α = b/r Hyperboic Functions sinh x = 2 (ex e x cosh x = 2 (ex + e x cosh 2 x sinh 2 x = sech 2 x + tanh 2 x = 2 sinh x cosh x = sinh 2x cosh 2x = 2 cosh 2 x = 2 sinh 2 x + [ sinh x = n x + ] ( + x 2, a x [ cosh x = n x + ] (x 2, x tanh x = ( + x 2 n, x x < coth x = ( x + 2 n, x x > MAS250 4 Continued
Dierentiation and Integration Function x n n x e x tan x cot x sec x cosecx sinh x cosh x tanh x coth x sechx Derivative nx n x e x sec 2 x cosec 2 x sec x tan x cosecx cot x cosh x sinh x sech 2 x cosech 2 x sechx tanh x cosechx cosechx coth x sin x ( x2 cos x ( x2 tan x + x 2 cot x + x 2 sinh x cosh x tanh x (x2 + (x2 x 2, x < coth x x 2, x > MAS250 5 Turn Over
Function Integra ( x a 2 + x 2 a tan a ( x a 2 x 2 a tanh a ( x sin a2 x 2 a ( x sinh a2 + x 2 a ( x cosh x2 a 2 a Dierentiation and Integration Formuae d(uv = udv + v du d ( u = v du udv v v 2 b a uv = [u (integra of v] b a Partia Dierentiation Chain Rue b a du (integra of v. Suppose that z = f(x, y and that x and y are functions of t, i.e., x = x(t, y = y(t. Then dz dt = z x dt + z dy y dt 2. Suppose that z = f(x, y and that x and y are functions of the variabes r and s, i.e., x = x(r, s, y = y(r, s. Then z r = z x x r + z y y r, z s = z x x s + z y y s MAS250 6 Continued
First-Order Dierentia Equations. Direct Integration dy = f(x y = f(x + C 2. Separation of Variabes dy = f(xg(y dy g(y = f(x 3. Homogeneous Equations make the substitution y = zx to give 4. Linear Equations dy ( y = f x z + x dz = f(z dy + P (xy = Q(x mutipy both sides by the integrating factor e P (x to give d ( ye P (x = Q(xe P (x MAS250 7 Turn Over
The Second-Order Dierentia Equation where a, b, and c are constants. Genera soution is a d2 y 2 + bdy + cy = f(x y = Compementary Function + Particuar Integra The soution, y c, is given by (i y c = Ae m x + Be m 2x, if m and m 2 rea and dierent, (ii y c = e mx (A + Bx, if m and m 2 rea and equa (m = m 2 = m, (iii y c = e px (A cos qx+b sin qx, if m and m 2 are compex (m = p+iq, m 2 = p iq, where m and m 2 are the roots of the auxiiary equation am 2 + bm + c = 0 Particuar Integra, y p f(x = Ax 2 + Bx + C y p = ax 2 + bx + c f(x = Ae kx y p = ae kx when k is not one of the roots of the auxiiary equation f(x = Ae kx y p = axe kx when k is one of the roots of the auxiiary equation f(x = A cos mx + B sin mx y p = a cos mx + b sin mx when sin mx or cos mx is not part of the compementary function f(x = A cos mx + B sin mx y p = x(a cos mx + b sin mx when sin mx or cos mx is part of the compementary function MAS250 8 Continued
Fourier Series Suppose that f(x is dened on the interva x. The Fourier series for f(x is given by f(x = a 0 2 + ( a n cos nπx + b n sin nπx, where a n = b n = n= f(x cos nπx f(x sin nπx, n = 0,, 2,...,, n = 0,, 2,.... On the interva 0 x the Fourier cosine series for f(x is f(x = a 0 2 + a n cos nπx and the Fourier sine series is n=, a n = 2 0 f(x cos nπx f(x = n= b n sin nπx, b n = 2 0 f(x sin nπx. Vector Cacuus The gradient of the scaar ed ϕ(x, y, z is given by ϕ = ( ϕ x, ϕ y, ϕ z The divergence of a vector ed u(x, y, z = (u, v, w is given by u = u x + v y + w z The cur of a vector ed u(x, y, z = (u, v, w is given by i j k u = x y z u v w. The Lapacian 2 is given by 2 = 2 x 2 + 2 y 2 + 2 z 2 MAS250 9 Turn Over
Statistics For data vaues (x, y, (x 2, y 2,..., (x n, y n Means x = n x i etc. n i= Variances s 2 x = n n (x i x 2 = n i= s x is standard deviation n (x 2 i x 2 etc. i= Covariance cov(x, y = n n (x i x(y i ȳ = n i= n (x i y i xȳ i= Correation coecient r = cov(x, y s x s y Linear regression by east squares The east squares t to the inear reationship y = a + b(x x is given by a = ȳ, b = cov(x, y s 2 x The corresponding mean square residua is s 2 y( r 2. MAS250 0