erwachsenc6

erwachsenc6

Answered

2022-10-17

How to find the zeroes of the function S, below?
want to find the zeroes of S = 4 ψ 2
Where
ψ ˙ 1 = 2 ψ 2
ψ ˙ 2 = 2 ψ 1

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Answer & Explanation

wlanauee

wlanauee

Expert

2022-10-18Added 17 answers

When you have
ψ ˙ 1 = 2 ψ 2
and
ψ ˙ 2 = 2 ψ 1 ,
you can re-write this as
( ψ ˙ 1 ψ ˙ 2 ) = ( 0 2 2 0 ) ( ψ 1 ψ 2 ) .
Call the matrix above A. A takes the general form of a matrix M defined as
M = ( σ ω ω σ ) ,
and the exponential for such matrices takes the form
exp ( M t ) = e σ t ( cos ( ω t ) sin ( ω t ) sin ( ω t ) cos ( ω t ) ) .
Using the values of σ = 0 and ω = 2 we have in A above, we get
exp ( A t ) = ( cos ( 2 t ) sin ( 2 t ) sin ( 2 t ) cos ( 2 t ) ) .
Then solving for ψ 1 and ψ 2 gives
( ψ 1 ( t ) ψ 2 ( t ) ) = ( cos ( 2 t ) sin ( 2 t ) sin ( 2 t ) cos ( 2 t ) ) ( ψ 1 ( t 0 ) ψ 2 ( t 0 ) ) .
From this we get that
S = 4 sin ( 2 t ) ψ 1 ( t 0 ) + 4 cos ( 2 t ) ψ 2 ( t 0 ) .
You can plug in any initial condition, ( ψ 1 ( t 0 ) , ψ 2 ( t 0 ) ) T of interest and solve for the zeros of S from there.

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