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Example
Paper 1 – Coupled Differential Equations
12
Consider the coupled differential equations
matrix equation
two 2� 1 matrices .
X � MX , where M is a 2� 2 matrix , and
Let �
1
and �
2 be the eigenvalues of M , where �1 � �2.
�dx
�2x�15y
� dt ��
. It can be expressed by a dy
� �x� 4y
�� 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 �
1
and �
2
.
Let v
1
and v
2 be the eigenvectors of M corresponding to �
1
and �
2 respectively .
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( c ) Write down v
1
and v
2
.
( d ) Hence , write down the general solution for X .
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Solution
( a )
2 �� �15 det ( M��I ) � ( M1 ) for valid approach � 1 4 � � det ( M��I ) � ( 2 ��)( 4 ��) �( �15 )( �1 )
2 det ( M��I
) � 8� 2� � 4� � � �15
2 det ( �� ) � � �6� �7
M I A1
( b ) �1 �� 1, �2 � 7 A2
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