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16 Integration 3 – Applications

Now when x 1,

dy
8
dx

1 8 e 5 4 12 d
1 d e 5
dy
e 5x 4x3 12x e 5
dx
The final integration gives:
So

˛

冮 1e

5x

y

4x3 12x e 5 2 dx
˛

e 5x
x4 6x2 e 5x f
5
When x 0, y 0 gives:
1 y

˛

˛

˛

1
0 f
5
1
5

1f

e 5x
1
x4 6x2 e 5x
5
5

Therefore y

˛

˛

˛

Exercise 1
Find the general solutions of these differential equations.
1

dy
x2 sin x
dx

2

dy
13x 72 4
dx

3

dy
1
2x 11 x2 2 2
dx

4

dy
x sin x
dx

5

dy
cos x

dx
1 sin x

6

dy
2kx
xe 3
dx

˛

dy
dy
5x

sin2 2x
8
2
dx
dx
21 15x
d2y
d3y
2

sec
x
x ln x
10
11
dx2
dx3
7

9

˛

˛

d2y
˛

dx2

˛

1

13x 22 2

˛

˛

˛

˛

12

˛

d4y
˛

dx4

x cos x

˛

Find the particular solutions of these differential equations.
13

dy
4x
2
given that when x 2, y 0
dx
4x 3
˛

14
15

dy
p
p
3 sin¢4x ≤ given that when x , y 2
dx
2
4
˛

d2y

1 dy
12x 12 4 given that when x ,
2 and that when
2 dx
dx
˛

2

˛

x 1, y 4
16

d2y
˛

dx

˛

2



2
p dy
3 and that when
given that when x ,
2
4 dx
1 x
˛

x 0, y 5

451