Method of Undetermined Coefficients - Examples

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

\begin{align} y_p &= t \\ &= t \\ &= t \\ &= (A_0 t^2 + A_1 t) \cos{t} + (B_0 t^2 + B_1 t) \sin{t} .\\ \end{align}

Substituting yp into the differential equation, we have the identity

\begin{align}t \cos{t} &= y_p'' + y_p \\ &= '' \\ &\quad + \\ &= \\ &\quad + \\ &\quad + \\ &= \cos{t} + \sin{t}. \\ \end{align}

Comparing both sides, we have

\begin{array}{rrrrl} &&4B_0&&=1\\ 2A_0 &&& + 2B_1 &= 0 \\ -4A_0 &&&& = 0 \\ &-2A_1 &+ 2B_0 && = 0 \\ \end{array}

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:

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