92.236 Engineering Differential Equations
The Method of Undetermined Coefficients
The method of undetermined coefficients can be used to find a particular solution ypof a non-
homogeneous linear d.e. if the d.e. has constant coefficients and the nonhomogeneous term is a
polynomial, an exponential, a sine or a cosine, or a sum or product of these.
If the d.e. has variable coefficients and/or the nonhomogeneous term is something
other than a polynomial, exponential, sine, cosine, or sum or product of these, you
must use another method (e.g. variation of parameters).
1. If the nonhomogeneous term is a polynomial, try guessing a polynomial of the same degree
for yp.
Example. y00
−3y0+2y=4x
Nonhomogeneous term: 4x(polynomial of degree 1).
Guess: yp=Ax +B(a polynomial of degree 1).
Note: You need the constant term Bin your guess for ypeven though there is no
constant in the nonhomogeneous term.
2. If the nonhomogeneous term is an exponential function, try guessing an exponential function
of the same form for yp.
Example. y00
−3y0+2y=6e−x
Nonhomogeneous term: 6e−x(an exponential function).
Guess: yp=Ae−x(an exponential function of the same form).
3. If the nonhomogeneous term is a sine or a cosine, try guessing a combination of sine and
cosine of the same angular frequency for yp.
Example. y00
−3y0+2y= 10 sin(2x)
Nonhomogeneous term: 10 sin(2x) (sine function with angular frequency 2).
Guess: yp=Asin(2x)+Bcos(2x) (combination of sine and cosine w. angular frequency 2).
Note: You need the cosine term Bcos(2x)in your guess for ypeven though there
is no cosine in the nonhomogeneous term.
4. If the nonhomogeneous term is a sum of a polynomial, an exponential function, and/or a sine
or cosine, try guessing a sum of these functions for yp.
Example. y00
−3y0+2y=4x+ 10 sin(2x)
Nonhomogeneous term: 4x+ 10 sin(2x) (sum of a polynomial of degree 1 and a sine with
angular frequency 2.)
Guess: yp=Ax +B+Csin(2x)+Dcos(2x)
5. If the nonhomogeneous term is a product of a polynomial, an exponential function, and/or a
sine or cosine, try guessing a product of these functions for yp.
Example. y00
−3y0+2y=36xe−x
Nonhomogeneous term: 36xe−x(product of a polynomial of degree 1 and an exponential.)
Guess: yp=(Ax +B)e−x
Note: You need the Be−xterm in your guess for ypeven though there is no such
term in the nonhomogeneous term.
OVER
