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MATH2001/7000 ASSIGNMENT 1

SEMESTER 2 2023

Due at 5:00pm 18 August. Marks for each question are shown. Total marks: 25

Submit your assignment online via the assignment 1 Gradescope submission link in Black-board.

(1) (4 marks.) You are given that the ODE

2xy + (y2 − 3x2 )y' = 0

is almost exact, i.e. there exists an “integrating factor” h(y) such that

2xyh + (y2 − 3x2 )h y' = 0

is exact. Find the integration factor h(y) and use your result to obtain the general solution of the almost exact ODE.

(2) (4 marks.) Find the general solution of the ODE

y'' + 2y' + 2y = e−x (1 + cos x).

(3) (5 marks.) Let β = {e1, e2, e3} be a basis of P2(R), where e1 = x − 1, e2 = x + 1, e3 = x 2 + 3. Find the coordinate vector of v = x 2 + x relative to β and show that {e1, e2, v} is a basis for P2(R).

(4) (4 marks.) Let the solution space U of the system of equations

2x1 + x2 + 3x3 − x4 + x5 = 0,

3x1 + 2x2 − 2x4 + x5 = 0,

3x1 + x2 + 9x3 − x4 + x5 = 0,

be the subspace of R 5 endowed with the usual dot product. Find a basis for U and a basis for the orthogonal complement U ⊥.

(5) (8 marks.) Let and define

(a) Show h A, Bi is an inner product of M2,2(R).

(b) Find an orthonormal basis for the subspace U of M2,2(R) spanned by