How to solve this system of ODE's? [ <mtable rowspacing="4pt" columnspacing="1em">

Dale Tate 2022-06-24 Answered
How to solve this system of ODE's?
[ x ˙ 1 x ˙ 2 ] = [ cos t sin t sin t cos t ] [ x 1 x 2 ]
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Answers (1)

Angelo Murray
Answered 2022-06-25 Author has 23 answers
This is a linear system of the form
x = A ( t ) x ,
where x R 2 , A ( t ) C ( R ; R 2 × 2 ), and most important
(1) A ( s ) A ( t ) = A ( t ) A ( s ) .
Satisfaction of ( 1 ) implies that the solution of
x = A ( t ) x , x ( 0 ) = ξ 0 ,
In our case
A ( t ) = ( cos t sin t sin t cos t ) ,
then
0 t A ( s ) d s = ( sin t cos t 1 1 cos t sin t ) .
Next we use use the fact that
exp ( a b b a ) = ( e a cos b e a sin b e a sin b e a cos b ) ,
and finally we obtain that
x ( t ) = ( e sin t cos ( 1 cos t ) e sin t sin ( 1 cos t ) e sin t sin ( 1 cos t ) e sin t cos ( 1 cos t ) ) ξ 0 .
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