is where 21,2 are the solutions to the auxiliary equation ay" + by' + cy=0 y = A₁e¹¹x + A₂¹2x aλ² + bλ + c = 0. In the case of complex roots λ = piw, we instead used the solution y = ex (C cos wx + D sin @x) (a) To show that the two solutions are equivalent, substitute the complex roots into the first expression, and compare it to the second, identifying formulae for C and D in terms of A₁ and A2. (b) Now find formulae for A₁ and A₂ in terms of C and D. If C and D are real (so that the solution is real), what is the relation between A₁ and A₂?

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
icon
Related questions
Question
100%

the the solution of a 2nd order linear ODE with constant coefficients (see picture for rest of question)

is
where 1,2 are the solutions to the auxiliary equation
ay" + by' + cy=0
y = A₁e¹¹x + A₂¹²x
aλ² + bλ + c = 0.
In the case of complex roots λ = piw, we instead used the solution
y = ex (C cos @x + D sin @x)
(a) To show that the two solutions are equivalent, substitute the complex roots into the first expression, and compare it to the second,
identifying formulae for C and D in terms of A₁ and A₂.
(b) Now find formulae for A₁ and A₂ in terms of C and D. If C and D are real (so that the solution is real), what is the relation between A₁
and A2?
Transcribed Image Text:is where 1,2 are the solutions to the auxiliary equation ay" + by' + cy=0 y = A₁e¹¹x + A₂¹²x aλ² + bλ + c = 0. In the case of complex roots λ = piw, we instead used the solution y = ex (C cos @x + D sin @x) (a) To show that the two solutions are equivalent, substitute the complex roots into the first expression, and compare it to the second, identifying formulae for C and D in terms of A₁ and A₂. (b) Now find formulae for A₁ and A₂ in terms of C and D. If C and D are real (so that the solution is real), what is the relation between A₁ and A2?
Expert Solution
steps

Step by step

Solved in 3 steps

Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning