Please solve the 6 questions of this assignment. I would like to receive the first milestone in two days, and the completed one on 16/12. But that would be great if both can be done as soon as possible.
School of Mathematics and Physics, UQ MATH2001/7000, Assignment 1, Summer 2020 (1) Give an explicit solution to the initial value problem y2 + 2xy + 2x− 1 + (2xy + x2)dy dx = 0, y(1) = 0. Show all working. (2) (a) Let A,B ∈ R be arbitrary constants. Verify that the function f(x) = A(1+x)+Bex is a solution to the differential equation xy′′ − (1 + x)y′ + y = 0. (b) Find the general solution to the differential equation xy′′ − (1 + x)y′ + y = x2e2x. Show all working. (3) Consider the matrix A = 1 4 5 6 9 3 −2 1 4 −1 −1 0 −1 −2 −1 2 3 5 7 8 (a) Give a basis for the row space of A. Show all working. (b) Give a basis for the null space of A. Show all working. (4) Let p(x), q(x) be continuous functions. Consider the ODE y′′ + py′ + qy = 0. (a) From lectures, recall that the set of all solutions to this ODE gives rise to a vector space, V . Show that V is an inner product space with inner product 〈f, g〉 = f(0)g(0) + f ′(0)g′(0). (b) Show that {cosh(x), sinh(x)} is an orthonormal basis with respect to the inner product of part (a), for the inner product space of solutions to the ODE y′′ − y = 0. (5) Consider the following data points: (−1,−14), (0,−5), (1,−4), (2, 1), (3, 23). (a) Find the least squares cubic fit y = a0 + a1x+ a2x 2 + a3x 3 to the data points. (b) Use a computational plotting tool (e.g. MATLAB) to plot the data points and the fitted curve on the same axes. (6) Consider the matrices A = 1 1 11 1 1 1 1 1 , B = 1 0 −10 0 0 −1 0 1 , C = 1 −2 1−2 4 −2 1 −2 1 , and M = 1 + α 1 1− α1 1 1 1− α 1 1 + α , α ∈ R. You are given that the matrices A,B and C satisfy AB = BA = AC = CA = BC = CB = 0, A2 = 3A, B2 = 2B, C2 = 6C. Find an invertible matrix P and a diagonal matrix D such that M = PDP−1. Show all working. Each question marked out of 3. • Mark of 0: No relevant answer submitted, or no strategy present in the submission. • Mark of 1: The submission has some relevance, but does not demonstrate deep under- standing or sound mathematical technique. • Mark of 2: Correct approach, but needs to fine-tune some aspects of the calculations. • Mark of 3: Demonstrated a good understanding of the topic and techniques involved, with well-executed calculations. Q1: Q2(a): Q3(a): Q4(a): Q5(a): Q6: Q2(b): Q3(b): Q4(b): Q5(b): Total (out of 30):