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{\Large\bf The Method of Undetermined Coefficients}

\vspace{.2in}

The method of undetermined coefficients is a technique
for determining the particular solution to linear
constant-coefficient differential equations
$$
y''(t)+py'(t)+qy(t)=f(t)
$$
for certain types of nonhomogeneous terms f(t).

This section will cover:

\begin{itemize}
\item f(t)=exp(at) 
\item f(t)=polynomial
\item f(t)=sine or cosine
\item f(t)=sum of various terms
\item Modified Guesses
\item Summary
\end{itemize}

\vspace{.3in}

{\bf Nonhomogeneous term f(t)=exp(at)}

\vspace{.2in}

Consider the example:

$$
y''+3y'+2y=e^{3t}
$$

We guess that the particular solution has the form
y_p(t)=Aexp(3t). Here A is a unknown constant. In this
case we have y'_p=3Aexp(3t) and  y''_p=9Aexp(3t).
Substituting the expressions for y_p, y'_p, and y''_p
into the differential equation, we obtain

$$
9Ae^{3t}+3(3e^{3t})+2e^{3t}=e^{3t}
$$

The LHS of this last expression is 20Ae^{3t}. Comparing
the two sides of the differential equation, we conclude
that A=1/20. 

The approach for this example is standard for a
constant-coefficient differential equations with
exponential nonhomogeneous term. If the nonhomogeneous 
term is constant times exp(at), then the initial guess
should be Aexp(at), where A is an unknown coefficient
to be determined. This guess may need to be modified.

\vspace{.3in}

{\bf Nonhomogeneous term f(t)=polynomial}

\vspace{.2in}

Consider the example:

$$
y''+3y'+2y=t+1
$$

We guess that the particular solution has the form
y_p(t)=At+B, where A and B are unknowns. In this
case we have y_p=A and y_p=0. Substituting the
expressions for y_p, y'_p, and y''_p into the
differential equation, we obtain

$$
0+3A+2(At+B)=t+1
$$

Rearranging the LHS of this equation, we obtain

$$
2At+(3A+B)=t+1
$$

Comparing the coefficients of t on both sides of the
equation, we conclude that 2A=1. Comparing the
constant terms, we conclude that 3A+B=1. The first
equation implies A=1/2. This fact and the second
equation imply that B=-1/2.

If the nonhomogeneous term is a polynomial of degree n,
then an initial guess for the particular solution should
be

$$
y_p=A_nt^n+A_{n-1}t^{n-1}+A_{n-2}t^{n-2}+\cdots+A_1t+A_0
$$

This guess may need to be modified. 

\vspace{.3in}

{\bf Nonhomogeneous term f(t)=sine or cosine}

\vspace{.2in}

Consider the example:

$$
y''+3y'+2y=\sin 3t
$$

We guess that the particular solution is
y_p=Asin(3t)+Bcos(3t). Note that we have both sine
and cosine terms. In this case we have

$$
y'_p=3A\cos 3t-3B\sin 3t \hspace{.2in} y''_p=-9A\sin 3t-9B\cos 3t
$$

Substituting y_p, y'_p, and y''_p into the differential
equation we obtain:

$$
-9A\sin 3t-9B\cos 3t+3(3A\cos 3t-3B\sin 3t)+2(A\sin 3t+B\cos 3t)=
\sin 3t
$$

Rearranging the LHS of this equation, we obtain: 

$$
(-7A-9B)\sin 3t+(9A-7B)\cos 3t=\sin 3t
$$

Comparing the coefficents of sin(3t) on both sides of
the equation, we conclude -7A-9B=1. The RHS has no
cosine terms. This means that the coefficient of
cosine on the LHS must be 0 -- that is, 9A-7B=0.
Solving these equations we conclude
A=-7/130 and B=-9/130.

In general if the nonhomogeneous term is
of the form a*sin(ct)+b*cos(ct). Then an 
guess for the particular solution is
y_p=Asin(ct)+Bcos(ct).

\vspace{.3in}

{\bf f(t)=sum of various terms}

\vspace{.2in}

Consider the differential equation 

$$
y''+3y'+2y=e^{3t}+\sin 3t
$$

If the nonhomogeneous term is a sum of two terms, then
the particular solution is y_p=y_p1 + y_p2, where
y_p1 is a particular solution of y''+3y'+2y=exp(3t)
and y_p2 is a particular solution y''+3y'+2y=\sin(3t) 

\vspace{.3in}

{\bf Modified Guesses}

\vspace{.2in}

Consider the example:

$$
y''+3y'+2y=e^{-t}
$$

According to the arguments above, the guess for
the particular solution is y_p=Aexp(-t). We
have y'_p=-Aexp(-t) and y''_p=Aexp(-t). Substituting
these expressions into the ode, we have

$$
Ae^{-t}-3Ae^{-t}+2Ae^{-t}=e^{-t}
$$

\end{document}
