Your Practice Set – Applications and Interpretation for IBDP Mathematics SE Production Limited 200 ( c ) 1 5 1 � � � � � � � v , 2 3 1 � � � � � � � � v A2 [ 2 ] ( d ) 7 5 3 1 1 t t Ae Be � � � � � � � � � � � � � � � � X A2 [ 2 ] 1 . Consider the coupled differential equations d 5 6 d d 4 3 d x x y t y x y t � � � � �� � � �� . It can be expressed by a matrix equation � X MX , where M is a 2 2 � matrix , and d d d d x t y t � � � � � � � � � � � � � X and x y � � � � � � � X are two 2 1 � matrices . Let 1 � and 2 � be the eigenvalues of M , where 1 2 � � � . ( a ) Find det ( ) �� M I , giving the answer in terms of � . [ 2 ] ( b ) Hence , write down the values of 1 � and 2 � . [ 2 ] Let 1 v and 2 v be the eigenvectors of M corresponding to 1 � and 2 � respectively . ( c ) Write down 1 v and 2 v . [ 2 ] ( d ) Hence , write down the general solution for X . [ 2 ] 2 . Consider the coupled differential equations d 2 15 d d 2 d x x y t y x y t � � � � �� � � �� . It can be expressed by a matrix equation � X MX , where M is a 2 2 � matrix , and d d d d x t y t � � � � � � � � � � � � � X and x y � � � � � � � X are two 2 1 � matrices . Exercise 55