ODE Lecture Notes
Section 3.5
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Section 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients
Big idea: One way to solve a nonhomogeneous linear second-order differential equations with
constant coefficients ay  by  cy  g  t  is to “guess” a particular solution that has the same
form as g(t), then work out the values of the coefficients that make that guess work.
Big skill: You should be able to solve nonhomogeneous second order differential equations with
constant coefficients using the method of “undetermined coefficients,” which is to say that you
should be able to find the homogeneous solution, the particular solution and its coefficient, and
then add them together for the general solution.
Recall that a nonhomogeneous equation is of the form L  y   y  p  t  y  q  t  y  g  t  .
Practice:
1. Verify that y  t   e2t  e3t  sin  4t  is a solution of
y  5 y  6 y  20cos  4t  10sin  4t  .
ODE Lecture Notes
Section 3.5
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The general solution to the nonhomogeneous equation will be the general solution of the
corresponding homogeneous equation (i.e., L  y   0 ) plus one or more additional terms called
the particular solution. This is because if yh(t) is the solution to ay  by  cy  0 , and yp(t) is
the solution to ay  by  cy  g  t  , then y  t   yh  t   y p  t  is the most general solution to the
nonhomogeneous equation because
ay  t   by  t   cy  t    ayh  t   byh  t   cyh  t     ay p  t   by p t   cy p t  
 0  g t 
 g t 
Theorem 3.5.1: The Difference of Nonhomogeneous Solutions Is the Homogeneous Solution
If Y1 and Y2 are two solutions of the nonhomogeneous equation
L  y   y  p  t  y  q t  y  g t  , then their difference Y1  Y2 is a solution of the corresponding
homogeneous solution L  y   y  p  t  y  q  t  y  0 . If, in addition, y1 and y2 are a
fundamental set of solutions of the homogeneous equation, then
Y1  Y2  c1 y1  t   c2 y2 t  ,
for certain constants c1 and c2.
Theorem 3.5.2: General Solution of a Nonhomogeneous Equation
The general solution of the nonhomogeneous equation L  y   y  p  t  y  q  t  y  g  t  , can
be written in the form y    t   c1 y1  t   c2 y2 t   Y t  where y1 and y2 are a fundamental set
of solutions of the corresponding homogeneous equation, c1 and c2, are arbitrary constants, and Y
is some specific solution of the nonhomogeneous equation.
Method of Undetermined Coefficients:
 When g(t) is an exponential, sinusoid, or polynomial, make a guess for the particular
solution that is the same exponential, sinusoids of the same frequency, or polynomial of
the same degree, except with arbitrary coefficients that you determine by substituting
your guess into the nonhomogeneous equation.
 This technique works because derivatives of these types of functions result in the same
type of function.
 If the form for the particular solution replicates any terms of the homogeneous solution,
then factors of t must be applied until there is no replication.
ODE Lecture Notes
Section 3.5
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Forms for the Particular Solution of ay  by  cy  g  t  for Assorted g(t):
g t   Pn t  (i.e., a polynomial of degree n)
y p t   Qn t  (i.e., a polynomial of degree n,
but with different coefficients than Pn)
Exception #1: r = 0 is a root of the
homogeneous equation 
y p t   t  Qn t 
g t   A cost   B sin t 
Exception #2: r = 0 is a double root of the
homogeneous equation 
y p t   t 2  Qn t 
y p t   C1 cost   C 2 sin t 
Exception: r = ± are roots of the
homogeneous equation 
y p t   C1t cost   C 2 t sin t 
g t   A  e kt
y p t   C1e kt
Exception #1: r = k is a root of the
homogeneous equation 
y p t   C1tekt
Exception #2: r = k is a double root of the
homogeneous equation 
y p t   C1t 2 e kt
g  t   ekt  Pn  t 
y p  t   Qn  t  ekt (i.e., the product of a
polynomial of degree n, and an exponential).
Exception #1: r = k is a root of the
homogeneous equation 
y p  t   tQn  t  ekt
Exception #2: r = k is a double root of the
homogeneous equation 
y p  t   t 2Qn  t  ekt
g  t   ekt  A cos t   B sin t  
g  t   Pn  t   cos t   sin t  
y p  t   ekt  C1 cos t   C2 sin t  
Exception #1: r = k + i is a root of the
homogeneous equation 
y p  t   tekt  C1 cos t   C2 sin t  
ODE Lecture Notes
Section 3.5
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Practice:
2. Find the general solution for y   2t  3 by direct integration and by the method of
undetermined coefficients.
ODE Lecture Notes
Section 3.5
3. Solve the IVP y  t   y  t   2 y  t   2e3t , y(0) = 1, y’(0) = 1.
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ODE Lecture Notes
Section 3.5
4. Solve the IVP y  t   y  t   2 y t   2sin t  , y(0) = 1, y’(0) = 1.
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ODE Lecture Notes
Section 3.5
5. Solve the IVP y  t   y  t   2 y  t   2et sin t  , y(0) = 1, y’(0) = 1.
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ODE Lecture Notes
Section 3.5
6. Solve the IVP y  t   y  t   2 y  t   2et , y(0) = 1, y’(0) = 1.
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