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If the roots of the auxiliary equation of a second order linear homogeous ODE are k1 >0 and -k2 < 0, then the solution is x(t) = Ae*it + Be-k2t. For most choice of initial conditions x(0) = x0, i(0) = yo we will have that r(t) → ±0 as t → 0. However, there are some special initial conditions for which x(t) → 0 as t → 0. Find the relationship between ro and yo that ensures this.

If the roots of the auxiliary equation of a second order linear homogeous ODE are k1 >0 and -k2 < 0, then the solution is x(t) = Ae*it + Be-k2t. For most choice of initial conditions x(0) = x0, i(0) = yo we will have that r(t) → ±0 as t → 0. However, there are some special initial conditions for which x(t) → 0 as t → 0. Find the relationship between ro and yo that ensures this.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 15EQ
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Problem 9** If the roots of the auxiliary equation of a second order linear homogeous ODE are ki > 0 and -k2 < 0, then the solution is x(t) = Ae*it + Be¯k2t. For most choice of initial conditions x(0) = x0, i(0) = Yo we will have that x(t) → t0 as t → 0. However, there are some special initial conditions for which x(t) → 0 as t → 0. Find the relationship between xo and yo that ensures this.
Transcribed Image Text:Problem 9** If the roots of the auxiliary equation of a second order linear homogeous ODE are ki > 0 and -k2 < 0, then the solution is x(t) = Ae*it + Be¯k2t. For most choice of initial conditions x(0) = x0, i(0) = Yo we will have that x(t) → t0 as t → 0. However, there are some special initial conditions for which x(t) → 0 as t → 0. Find the relationship between xo and yo that ensures this.
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