Your Practice Set – Applications and Interpretation for IBDP Mathematics
Exercise 81
1 . This question aims at investigate the relationship between the second-order differential equation and the eigenvalues of the corresponding coupled system of first-order differential equations .
Consider the coupled system
�dv
�4v�12y
�dx ��
. dy
� v
�� dx
( a ) Show that the coupled system can be expressed as
2 d y dy � 4 �12y � 0 .
2 dx dx
( b ) The coupled system can be expressed by a matrix equation Y � MY , where M is
a 2� 2 matrix , and
( 1 ) Write down M .
�dv �
� dx �
Y � � � and dy � � �dx �
�v �
Y � � � are two 2� 1 matrices . �y �
Let �
1
and �
2 be the eigenvalues of M , where �1 � �2.
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( 2 ) Find the characteristic polynomial of M .
( 3 ) 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 .
( 4 ) Write down v
1
and v
2
.
( 5 ) Hence , write down the general solution for y .
( 6 ) By considering the expressions of d 2
y dx and d y
, show that the general
2 dx solution for y satisfies the differential equation
2 d y dy � 4 �12y � 0 .
2 dx dx
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