Examples
- Example 1
Find a particular integral of the equation
The right side t cos t has the form
with n=1, α=0, and β=1.
Since α + iβ = i is a simple root of the characteristic equation
we should try a particular integral of the form
Substituting yp into the differential equation, we have the identity
Comparing both sides, we have
which has the solution = 0, = 1/4, = 1/4, = 0. We then have a particular integral
- Example 2
Consider the following linear nonhomogeneous differential equation:
This is like the first example above, except that the nonhomogeneous part is not linearly independent to the general solution of the homogeneous part ; 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:
By substituting this function and its derivative into the differential equation, one can solve for A:
So, the general solution to this differential equation is thus:
- Example 3
Find the general solution of the equation:
f(t), is a polynomial of degree 2, so we look for a solution using the same form,
- , where
Plugging this particular integral with constants A, B, and C into the original equation yields,
- , where
- and and
Replacing resulting constants,
To solve for the general solution,
where is the homogeneous solution, therefore, the general solution is:
Read more about this topic: Method Of Undetermined Coefficients
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