Analysis and Approaches for IBDP Maths Ebook 2 | Page 216

Your Practice Set – Analysis and Approaches for IBDP Mathematics
66
Paper 1 Section A – Separating Variables
Example
Solve the differential equation (32
dy
)
dx
cos
the answer in the form y � f ( x)
.
Solution
dy
(32 � ) � cos
dx
5 4 2
x x y
x
32 � x
4
2
sec ydy
� d
5
x
5 4 2
�x � x y , where 0
y � when x � 0 , giving
[8]
(M1) for valid approach
4
2 x
�sec ydy
� � dx
(A1) for correct approach
5
32 � x
5
Let u�32
� x . (M1) for substitution
du
4 1 4
5x du x dx
dx � � 5
�
2 1 1
��sec ydy
� � � du
(A1) for correct working
5 u
1
tan y � ln u � C
5
1
5
tan y � ln 32 � x � C
5
1
5
y arctan �
�
� � ln 32 � x � C �
A1
�5
�
�1
0 arctan � ln 32 0
�5
1
0 � ln 32 � C
5
5
� � �
�
C �
�
(M1) for substitution
1
C �� ln 32
(A1) for correct value
5
�1 5 1 �
� y� arctan � ln 32 � x � ln 32�
�5 5 �
5
�1 32 � x �
y � arctan
�
ln
5 32 �
�
�
A1
[8]
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