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Question 1 (a) Is the operator P(y) = y/ + 2ty linear? (Show workings) (b) Find the null space of the operator in (a) above.

Question 1 (a) Is the operator P(y) = y/ + 2ty linear? (Show workings) (b) Find the null space of the operator in (a) above.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 1 (a) Is the operator P(y) =y + 2ty linear? (Show workings) (b) Find the null space of the operator in (a) above. Question 2 Let P be a linear operator. Suppose that y is a particular solution to the equation P(y) = b. Prove that any solution to this equation can be written as y yo + Yı for yo an element of the null space of P. Question 3 Consider the differential equation dy dy - 2y = e'. dt dt2 (a) Write the equation in D-operator form, i.e. as P(D)(y) = e". (b) Find the general complementary function yo for the equation. (c) Find a particular integral yi for the equation. (d) Hence, write down the general solution y for the equation, and solve for the initial conditions y(0) = y'(0) = 0. Question 4 (a) Use D-operator methods to find the real-valued complementary function yo of the cquation (D? + 6D+ 13)y = 0. (b) Is the operator D? + 6D + 13 stable or not? Motivate your answer. If it is stable, is it underdarmped or overdamped? Question 5 Use D-operator methods to find the particular solution y, of the equation (D² + 6D + 13)y = e-3t sin(2t). Hint: s + 6s + 13 = (s + 3+2i)(s + 3 – 2i).
Transcribed Image Text:Question 1 (a) Is the operator P(y) =y + 2ty linear? (Show workings) (b) Find the null space of the operator in (a) above. Question 2 Let P be a linear operator. Suppose that y is a particular solution to the equation P(y) = b. Prove that any solution to this equation can be written as y yo + Yı for yo an element of the null space of P. Question 3 Consider the differential equation dy dy - 2y = e'. dt dt2 (a) Write the equation in D-operator form, i.e. as P(D)(y) = e". (b) Find the general complementary function yo for the equation. (c) Find a particular integral yi for the equation. (d) Hence, write down the general solution y for the equation, and solve for the initial conditions y(0) = y'(0) = 0. Question 4 (a) Use D-operator methods to find the real-valued complementary function yo of the cquation (D? + 6D+ 13)y = 0. (b) Is the operator D? + 6D + 13 stable or not? Motivate your answer. If it is stable, is it underdarmped or overdamped? Question 5 Use D-operator methods to find the particular solution y, of the equation (D² + 6D + 13)y = e-3t sin(2t). Hint: s + 6s + 13 = (s + 3+2i)(s + 3 – 2i).
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