Euler ’ s method with a step length of 0.25 is used to approximate the displacement of the particle at t � 1. It is given that initially the particle is at rest with displacement 1 m .
( b ) Find , when t � 0.25 , the approximate value of
( i ) v ;
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( ii ) x .
( c ) Hence , write down the approximate value of the displacement at
[ 4 ]
( i ) t � 0.5 ;
( ii ) t � 0.75 ;
( iii ) t � 1.
The system can be expressed by a matrix equation
and
�dv �
� dt �
X � � � and dx � � � dt
�
M , where �1 � �2.
[ 3 ] X � MX , where M is a 2� 2 matrix ,
�v �
X � � � are two 2� 1 matrices . Let �
1
and �
2 be the eigenvalues of
�x�
( 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 .
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[ 2 ]
( f ) Write down v
1
and v
2
. [ 2 ]
( g ) Find
( i ) the particular solution of x ;
( ii ) the displacement at t � 1.
( h ) Hence , calculate the percentage error for the approximated displacement in ( c )( ii ).
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