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?

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...

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