Your Practice Set – Applications and Interpretation for IBDP Mathematics
4 . Consider the coupled differential equations
matrix equation
two 2� 1 matrices .
X � MX , where M is a 2� 2 matrix , and
�dx
� �8x� y
� dt �� . It can be expressed by a dy �13x�2y
�� dt �dx �
� dt �
X � � � and dy � � � dt �
�x �
X � � � are �y �
( a ) Find det ( M��I ) , giving the answer in terms of � .
( b ) Hence , write down the values of the eigenvalues of M .
Let v
1
and v
2 be the eigenvectors of M .
[ 2 ]
[ 2 ]
( c ) Write down the general solution for X .
The phase portrait for the coupled differential equation is in the shape of spiral .
( d ) Describe the stability of the phase portrait .
[ 2 ]
[ 1 ]
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