Your Practice Set – Applications and Interpretation for IBDP Mathematics
59
Paper 2 – Miscellaneous Problems
Example
Under the influence of a magnetic field , the changes in the position ( x, y ) of a metal ball
can be modelled by the coupled differential equations
( 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.
�dx
�3x�2y
� dt �� . dy
�4x� y
�� dt
[ 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 .
[ 2 ]
[ 2 ]
( d ) Write down v
1
and v
2
.
The metal ball is initially at ( 2 , 5 ) .
[ 2 ]
( e ) Find the particular solution of
( i ) x ;
( ii ) y .
( f ) Hence , find the coordinates of the position of the metal ball at t � 0.5 .
[ 5 ]
[ 2 ]
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