Consider the differential equation u'' + 6u' + 9u = 0 : It has solutions u = e^-3t and u = t e^-3t. A.1. Compute the Wronskian Wr[e^-t; t e^-3t](t). How can we know that Wr[e^-3t; t e^-3t](t) is proportional to e^6t before we compute it? (Hint: Abel.) A.2. Why are e^-3t and t e^-3t linearly independent functions? A.3. Find the natural fundamental set of solutions to the equation associated with t = 0.
Consider the differential equation u'' + 6u' + 9u = 0 : It has solutions u = e^-3t and u = t e^-3t. A.1. Compute the Wronskian Wr[e^-t; t e^-3t](t). How can we know that Wr[e^-3t; t e^-3t](t) is proportional to e^6t before we compute it? (Hint: Abel.) A.2. Why are e^-3t and t e^-3t linearly independent functions? A.3. Find the natural fundamental set of solutions to the equation associated with t = 0.
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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Consider the differential equation
u'' + 6u' + 9u = 0 : It has solutions u = e^-3t and u = t e^-3t.
A.1. Compute the Wronskian Wr[e^-t; t e^-3t](t). How can we know that Wr[e^-3t; t e^-3t](t) is proportional to e^6t before we compute it? (Hint: Abel.)
A.2. Why are e^-3t and t e^-3t linearly independent functions?
A.3. Find the natural fundamental set of solutions to the equation associated with t = 0.
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