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MATH 371 - Summer 2008
Notes on Undetermined Coefficients
Undetermined Coefficients
So far, we have learned to solve second order linear homogeneous differential equations.
We now introduce a method for solving nonhomogeneous second order linear differential equations. That is, equations of the form y + p(x)y + q(x)y = r(x) where r(x) is not necessarily 0.
We will be considering the special case where the coefficients are constant (so p(x) = b and q(x) = c, say), and where r(x) is a sum of functions from the following list:
• keλx
• kxn (for n = 0, 1, 2, . . .; this includes constants, where n = 0)
• k cos(ωx) or k sin(ωx) (note the ω is the lowercase greek letter ω)
• keαx cos(ωx) or k αx sin(ωx)
So for example, any constant, any polynomial, 5e2x , − sin( x ), 3x2 + e−2x + ex cos(4x),
2
etc. would all be valid r(x).
The Main Idea
While method we will discuss is called the Method of Undetermined Coefficients, we might as well just call it “educated guessing,” because that’s all it is.
We know something about which functions will have derivatives that look like those listed above. So we make an educated guess about what the solution should look like, throwing in some “undetermined coefficients” to compensate for the fact that we don’t quite know exactly how the answer should look. We plug this guess in for y on left side of the equation
(y +by +cy), and simplify that side. Now, equality with the right side of the equation (r(x)) lets us solve for the undetermined coefficents and replace them with the actual coefficients in our original guess.
Now, if our guess was good enough — that is, if the only part of our guess that wasn’t perfect was the coefficients, we’re done. We have the solution. If on the other hand, we made a bad guess, we probably ran into some trouble back there.
The Method
There are three simple rules to follow in the Method of Undetermined Coefficients.
1. The Basic Rule.
In the table below each Ci is an undetermined coefficient. If
Notes on Undetermined Coefficients
Undetermined Coefficients
So far, we have learned to solve second order linear homogeneous differential equations.
We now introduce a method for solving nonhomogeneous second order linear differential equations. That is, equations of the form y + p(x)y + q(x)y = r(x) where r(x) is not necessarily 0.
We will be considering the special case where the coefficients are constant (so p(x) = b and q(x) = c, say), and where r(x) is a sum of functions from the following list:
• keλx
• kxn (for n = 0, 1, 2, . . .; this includes constants, where n = 0)
• k cos(ωx) or k sin(ωx) (note the ω is the lowercase greek letter ω)
• keαx cos(ωx) or k αx sin(ωx)
So for example, any constant, any polynomial, 5e2x , − sin( x ), 3x2 + e−2x + ex cos(4x),
2
etc. would all be valid r(x).
The Main Idea
While method we will discuss is called the Method of Undetermined Coefficients, we might as well just call it “educated guessing,” because that’s all it is.
We know something about which functions will have derivatives that look like those listed above. So we make an educated guess about what the solution should look like, throwing in some “undetermined coefficients” to compensate for the fact that we don’t quite know exactly how the answer should look. We plug this guess in for y on left side of the equation
(y +by +cy), and simplify that side. Now, equality with the right side of the equation (r(x)) lets us solve for the undetermined coefficents and replace them with the actual coefficients in our original guess.
Now, if our guess was good enough — that is, if the only part of our guess that wasn’t perfect was the coefficients, we’re done. We have the solution. If on the other hand, we made a bad guess, we probably ran into some trouble back there.
The Method
There are three simple rules to follow in the Method of Undetermined Coefficients.
1. The Basic Rule.
In the table below each Ci is an undetermined coefficient. If