12 . A differential equation is given by x � 1.
2 6 dy � � , where dx x � 2 3 x x 6y x e
7 e y � when
2
( a ) ( i ) By using a suitable integrating factor , solve the differential equation y � f ( x)
.
2 6 dy � � , giving the answer in the form dx x � 2 3 x x 6y x e
( ii ) Hence , write down the exact value of y when x � 3.
( b ) Use Euler ’ s method with a step length of 0.1 to
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( i ) show that an approximation for the value of y when x � 1.1 is 3 10
7 e ;
( ii ) write down an approximation for the value of y when x � 3, giving the answer correct to 4 decimal places .
( c ) Explain why the approximation for the value of y in ( b )( ii ) is an underestimate of the exact value of y in ( a )( ii ).
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