7 . A particle moves in a straight line with velocity v ms
�1 and displacement with respect to the starting point O . By considering the rate of change of its velocity , the relationship between the variables can be modelled by the
differential equation
( a ) By using
2 d x dx � 7 �10x
� 0.
2 dt dt x m dx v � , express the differential equation in a coupled system . dt
[ 1 ]
The system can be expressed by a matrix equation
2� 2 matrix , and
�dv �
� dt
� X � � � and dx � � � dt
� be the eigenvalues of M , where �1 � �2.
X � MX , where M is a
�v
� X � � � are two 2� 1 matrices . Let �
1 and �
2
�x�
( 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 .
( d ) Write down v
1 and v
2
.
Initially , the particle is at O and its velocity is
1
3 ms � .
[ 2 ]
[ 2 ]
[ 2 ]
( e ) Find the particular solutions of the velocity and the displacement of the particle .
[ 5 ]
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