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
icon
Related questions
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).
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).
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,