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
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
![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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7136435d-2c0b-4a6a-bb14-2bf831d8826c%2F04f706f0-d194-49f5-9096-2283189a4ae5%2F7i8psih_processed.jpeg&w=3840&q=75)
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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