School of Mathematics and Physics, UQ
MATH2001, Assignment 1, Summer Semester, 2022-2023
Due at 2:00pm 22 December. Each question marked out of 10. Total marks: 50
Submit your assignment online via the Assignment 1 submission link in Blackboard.
(1) Consider the equation

cos y + 2e?x cos x
y

y ? = 2e?x sin x ?
sin y
.
y
(a) Show that it is not exact, but it becomes exact when multiplying by the
integrating factor = yex .
?
(b) Solve the exact equation subject to the initial condition y(?) = . Present
2
your solution as a relation defining y implicitly as a function of x.
Show all working.
(2) Consider the non-homogeneous ODE
x2 y ?? ? 3xy ? + 4y = x2 ln x,
x > 0.
(a) Show that y1 = x2 and y2 = x2 ln x are solutions to the corresponding homogeneous ODE x2 y ?? ? 3xy ? + 4y = 0.
(b) Find the general solution of the non-homogeneous ODE.
Show all working.
(3) Consider the inner product space P2 (R) with inner product
p0 + p1 x + p2 x2 , q0 + q1 x + q2 x2 = p0 q0 + p1 q1 + p2 q2 , p0 , p1 , p2 , q0 , q1 , q2 ? R.
Let U = {1 + 2x + x2 }.
(a) Find U ? .
(b) Determine an orthogonal basis for U ? .
Show all working.
(4) Consider the vector space R4 endowed with the inner product
(u1 , u2 , u3 , u4 ) , (v1 , v2 , v3 , v4 ) = u1 v1 + 3u2 v2 + u3 v3 + 2u4 v4 .
Let U = span{u1 = (?2, 1, 0, 1), u2 = (0, 1, 2, 3)} be a subspace of R4 .
(a) Use Gram-Schmidt procedure to construct an orthonormal basis for U .
(b) Find the orthogonal projection of v = (?1, 2, 5, 1) in U and U ? .
Show all working.
(5) Let P2 (R) have the inner product,
Z 1
?p, q? =
p(x)q(x) dx,
?p, q ? P2 (R).
0
Find the best approximation of f (x) = x2 + x3 by polynomials in P2 (R). Show all
working.

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