Apps. and Interpretation for IBDP Maths Ebook 2 | Page 206

Your Practice Set – Applications and Interpretation for IBDP Mathematics SE Production Limited 196 Consider the differential equation 5 4 d ( 32 ) 5 d y y x x x � � , where x , 0 y � . ( a ) Express d yy � as an indefinite integral of x . [ 2 ] ( b ) Hence , using the substitution 5 32 u x � � to find the expression of 2 1 2 y , giving the answer in terms of x . [ 3 ] It is given that 0 y � when 0 x � . ( c ) Find the expression of y . [ 3 ] ( a ) 5 4 d ( 32 ) 5 d y y x x x � � 4 5 5 d d ( 32 ) x y y x x � � ( M1 ) for valid approach 4 5 5 d d 32 x y y x x � � � � � A1 [ 2 ] ( b ) Let 5 32 u x � � . 4 4 d 5 d 5 d d u x u x x x � � � A1 1 d d y y u u � � � � ( A1 ) for correct working 2 1 ln 2 y u C � � 2 5 1 ln ( 32 ) 2 y x C � � � A1 [ 3 ] Example Solution 54 Paper 1 – Separating Variables

Your Practice Set – Applications and Interpretation for IBDP Mathematics SE Production Limited 196 Consider the differential equation 5 4 d ( 32 ) 5 d y y x x x � � , where x , 0 y � . ( a ) Express d yy � as an indefinite integral of x . [ 2 ] ( b ) Hence , using the substitution 5 32 u x � � to find the expression of 2 1 2 y , giving the answer in terms of x . [ 3 ] It is given that 0 y � when 0 x � . ( c ) Find the expression of y . [ 3 ] ( a ) 5 4 d ( 32 ) 5 d y y x x x � � 4 5 5 d d ( 32 ) x y y x x � � ( M1 ) for valid approach 4 5 5 d d 32 x y y x x � � � � � A1 [ 2 ] ( b ) Let 5 32 u x � � . 4 4 d 5 d 5 d d u x u x x x � � � A1 1 d d y y u u � � � � ( A1 ) for correct working 2 1 ln 2 y u C � � 2 5 1 ln ( 32 ) 2 y x C � � � A1 [ 3 ] Example Solution 54 Paper 1 – Separating Variables