6 . In an experiment , a metal ball moves with velocity v cms
�1 and displacement x cm 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
2 d x � 25x
.
2 dt
dx ( a ) By using v � , express the differential equation in a coupled system . dt
[ 1 ]
Euler ’ s method with a step length of 0.2 is used to approximate the displacement of the particle at t � 1. It is given that initially the particle is at rest with displacement one centimetre .
( b ) Find , when t � 0.2 , the approximate value of
( i ) v ;
( ii ) x .
( c ) Write down the approximate value of the displacement at
[ 4 ]
( i ) t � 0.4 ;
( ii ) t � 1;
( iii ) t � 2.6.
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�
[ 3 ]
( d ) Find det ( M��I ) , giving the answer in terms of � .
( e ) 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 .
[ 2 ]
[ 2 ]
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