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MATH2001/2901/7000, Assignment 1, Semester 1, 2024

Total marks: 24

Due 1pm Monday 18 March 2024

This Assignment contributes 13% towards your final grade. In the absence of an approved extension, assignments submitted after the due date will attract a penalty as outlined in the course profile. Prepare your assignment as a pdf file, either by typing it, writing on a tablet or by scanning/photographing your handwritten work. Ensure that your name, student number and tutorial group number appear on the first page of your submission. Check that your pdf file is legible and that the file size is not excessive. Files that are poorly scanned and/or illegible may not be marked. Upload your submission using the Gradescope assignment submission link in Blackboard. In the submission process, after uploading your file, you must allocate page number(s) to each question!

(1) (3 marks) Consider the initial value problem

Use the method of successive approximations (with φ0(x) = 0) to generate a sequence of functions {φ0(x), φ1(x), φ2(x),..., φn(x)} for an arbitrary value of n ∈ N. Your answer should ultimately provide a formula for φn(x), verified using induction. Show all working.

(2) (3 marks) Determine a solution to the initial value problem

Show all working.

(3) (3 marks) Determine a 3rd order, linear, homogeneous ODE with general solution given by

where A, B, C ∈ R are arbitrary constants. Show all working.

(4) (a) (3 marks) Let A, B ∈ R be arbitrary constants. Verify that the function f(x) = A(1 + x) + Bex is a solution to the differential equation

(b) (3 marks) Find the general solution to the di↵erential equation

Show all working.

(5) (3 marks) Consider the matrix

Determine a basis for the orthogonal complement (with respect to the Euclidean inner product) of the column space of A  Show all working.

(6) Consider the inner product space C[0, 1] with inner product

(a) (3 marks) Show that {, 6x − 4} is an orthonormal basis for the subset

(b) (3 marks) Find the real numbers a and b such that ||x2 − a − b(6x − 4)|| is minimised.

Show all working.

Each question marked out of 3. Your mark will be largely based on the calculations and discussion you provide.

• 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.