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In a Nut Shell:
You
need to identify the type of nonhomogeneous
differential equation
and the nonhomogeneous function, f(x), before deciding on a
method to find the particular
solution. Two types to consider are second order,
linear, ordinary differential equations
with constant
coefficients and second order ones with variable coefficients.
Use the Method Undetermined Coefficients for:
Type 1: Second
order, linear, ordinary differential equations with constant coefficients,
such as:
a d2y/dx2 + b dy/dx + c y
= f(x)
provided the functions,
f(x), are of the type given in the table below.
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n
A polynomial
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n
Sine functions
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n
Cosine functions
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n
Sine and Cosine
functions
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n
Exponential
functions
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n
Or products of these
functions
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If f(x) is not among
this table of options, then the method of variation of parameters
should be used to obtain
particular solutions.
Use the Method of Variation of Parameters for:
Type 2A: Second
order, linear, ordinary differential equations with constant coefficients
where f(x) involves
quotients or functions not shown in the above table.
Examples include: f(x) = sec x, f(x) = tan x, f(x) = 1/x, f(x) = x/(x2 + 1), etc.
Type 2B: Second
order, linear, ordinary differential equations with variable coefficients
such as:
x2 y''(x) + 2x
y'(x) ˗ y(x)
= f(x) x > 0
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