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A-Level Mathematics, differential equations, solutions
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Understanding Pure Mathematics
AJ Sadler and DWS Thorning (1987)
ISBN 0199142432
ISBN 0199142599
ISBN 9780199142439
Solutions to Exercise 20D (p. 515–516)
1(a)
( )
Differentiating (1):
( )
Rearranging (2):”
Substituting (2) into (1):
(
)
1(b)
( )
( )
Differentiating (2):
( )
From (1):
( )
Substituting (4) into (3):
1(c)
1(d)
( )
1(e)
( )
Differentiating (1):
(
) ( )
Rearranging (2):
( )
Substituting (3) into (2):
1(f)
( )
Differentiating (1):
( )
From (1):
( )
Substituting (3) into (2):
1(g)
( )
Differentiating (1):
Divide throughout by 2:
1(h)
( ) ( )
( )
Differentiating (2):
( )
From (1):
( )
Substituting (3) into (4):
(
)
( )
1(i)
( )
Differentiating (1):
2.
3.
∫
∫
Simplifying:
Multiply throughout by 2:
√
4.
∫ ∫
Simplifying:
Multiplying throughout by :
5.
∫ ∫
Simplifying:
Rearranging:
6.
∫ ∫
Multiplying throughout by 2:
√
7.
∫ ∫
∫
( )
8.
∫ ∫
Multiplying throughout by :
( )
9.
∫
∫
10.
( )
∫
∫
( ) (
)
( )
11.
∫ ∫
( )
12.
( )
∫
∫
( )
13.
∫ ∫
( )
14.
∫
∫
( )
( )
( )
( )
15.
( )
( )
∫
∫
( ) ∫
∫
( )
( )
( )
( )
( )
16.
( )
∫
∫
17.
( ) ( )
18.
∫ ∫
( )
19.
( )
∫ ∫
√
20.
∫ ∫
( )
√
√
From (1):
(
)
(
)
√
21.
∫ ∫
( )
22.
( )
∫
∫
( )
( )
( )
( )
23.
∫
∫
