IBHM 446 472 .pdf
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16 Integration 3 – Applications
In certain situations we may be asked to solve differential equations other than first order.
Example
d4y
˛
Solve the differential equation
dx4
cos x, giving the general solution.
˛
From basic integration:
d3y
˛
dx3
d3y
冮 cos x dx
˛
sin x k
dx3
Continuing to integrate:
1
˛
˛
d2y
cos x kx c
dx2
dy
kx2
sin x
cx d
dx
2
˛
If we were given the
boundary conditions then
the constants k, c, d, and e
could be evaluated.
˛
˛
y cos x
kx3
cx2
dx e
6
2
˛
˛
Example
d3y
25e5x 24x
dx3
d2y
dy
8, and when
given that when x 1, 2 5e5, when x 1,
dx
dx
Find the particular solution to the differential equation
˛
˛
˛
˛
x 0, y 0.
d3y
˛
dx3
d2y
25e5x 24x
˛
Hence
˛
dx2
d2y
˛
1
˛
dx2
冮 125e
5x
24x2 dx
5e5x 12x2 c
˛
˛
d2y
5e5
dx2
1 5e5 5e5 12 c
1 c 12
When x 1,
˛
˛
1
d2y
˛
dx2
5e5x 12x2 12
˛
˛
Integrating again gives:
冮
dy
15e5x 12x2 122 dx
dx
dy
e5x 4x3 12x d
1
dx
˛
˛
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