Exercise 59
1 . In a physics experiment , the changes in the position ( x, y ) of a particle can be modelled
by the coupled differential equations
�dx
� �x� 5y
� dt �� . dy
� x�7y
�� dt
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( a ) Write down the coordinates of the equilibrium point .
The system can be expressed by a matrix equation
and
�dx �
� dt �
X � � � and dy � � � dt
�
M , where �1 � �2.
[ 1 ] X � MX , where M is a 2� 2 matrix ,
�x �
X � � � are two 2� 1 matrices . Let �
1
and �
2 be the eigenvalues of
�y �
( b ) Find det ( M��I ) , giving the answer in terms of � .
( c ) Hence , write down the values of �
1
and �
2
.
Let v
1
and v
2 be the eigenvectors of M corresponding to �
1
and �
2 respectively .
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( d ) Write down v
1
and v
2
.
The particle is initially at ( 9,1 ) .
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( e ) Find the particular solution of
( i ) x ;
( ii ) y .
( f ) Hence , find the coordinates of the position of the particle at t � 1.
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