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Problem Statement: Find an
approximate solution to the first order, ordinary d.e.
dy/dx =
f [ (x,y(x) ] subject to the condition y(a) = b
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Strategy:
Use successive approximations (another name for Picard's Method)
c c
First Integrate dy =
f[x,y(x)] dx from a to c or ∫ dy =
∫ f [x,y(x)] dx
a a
where a is the initial
value of x and c is an arbitrary value.
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c
Then y(c)
˗ y(a) =
∫ f [x,y(x)] dx but y(a)
= b so
a
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c
y(c)
˗ b =
∫ f [x,y(x)] dx here x
is a dummy variable
a
(switch variable of integration to
t)
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x
for arbitrary x, y(x) =
b + ∫ f [ t,y(t)
] dt
a
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x
Successive steps are given by: yn+1(x) =
b + ∫ f [ t,yn(t)
] dt
start with yo = b
a
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Click here for an
example.
Click here to return to
the discussion on approximate methods.
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