Approach for finding solutions of nonhomogeneous ordinary differential equations
In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) in order to find the best possible form of the particular solution, an ansatz or 'guess' is made as to the appropriate form, which is then tested by differentiating the resulting equation. For complex equations, the annihilator method or variation of parameters is less time-consuming to perform.
Undetermined coefficients is not as general a method as variation of parameters, since it only works for differential equations that follow certain forms.
Description of the method.
Consider a linear non-homogeneous ordinary differential equation of the form
formula_1
where formula_2 denotes the i-th derivative of formula_3, and formula_4 denotes a function of formula_5.
The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met:
The method consists of finding the general homogeneous solution formula_12 for the complementary linear homogeneous differential equation
formula_13
and a particular integral formula_14 of the linear non-homogeneous ordinary differential equation based on formula_15. Then the general solution formula_3 to the linear non-homogeneous ordinary differential equation would be
formula_17
If formula_15 consists of the sum of two functions formula_19 and we say that formula_20 is the solution based on formula_21 and formula_22 the solution based on formula_23. Then, using a superposition principle, we can say that the particular integral formula_14 is
formula_25
Typical forms of the particular integral.
In order to find the particular integral, we need to 'guess' its form, with some coefficients left as variables to be solved for. This takes the form of the first derivative of the complementary function. Below is a table of some typical functions and the solution to guess for them.
If a term in the above particular integral for "y" appears in the homogeneous solution, it is necessary to multiply by a sufficiently large power of "x" in order to make the solution independent. If the function of "x" is a sum of terms in the above table, the particular integral can be guessed using a sum of the corresponding terms for "y".
Examples.
Example 1.
Find a particular integral of the equation
formula_26
The right side "t" cos "t" has the form
formula_27
with "n" = 2, "α" = 0, and "β" = 1.
Since "α" + "iβ" = "i" is "a simple root" of the characteristic equation
formula_28
we should try a particular integral of the form
formula_29
Substituting "y""p" into the differential equation, we have the identity
formula_30
Comparing both sides, we have
formula_31
which has the solution 
formula_32 
We then have a particular integral
formula_33
Example 2.
Consider the following linear nonhomogeneous differential equation:
 formula_34
This is like the first example above, except that the nonhomogeneous part (formula_35) is "not" linearly independent to the general solution of the homogeneous part (formula_36); as a result, we have to multiply our guess by a sufficiently large power of "x" to make it linearly independent.
Here our guess becomes:
formula_37
By substituting this function and its derivative into the differential equation, one can solve for "A":
formula_38
formula_39
formula_40
So, the general solution to this differential equation is:
formula_41
Example 3.
Find the general solution of the equation:
formula_42
formula_43 is a polynomial of degree 2, so we look for a solution using the same form,
formula_44 
Plugging this particular function into the original equation yields,
formula_45
formula_46
formula_47
which gives:
formula_48
Solving for constants we get:
formula_49
To solve for the general solution,
formula_50
where formula_12 is the homogeneous solution formula_52, therefore, the general solution is: 
formula_53
References.
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