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					ODE Lecture Notes                              Section 3.5                                    Page 1 of 5

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 value of the coefficient that makes 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.



The general solution to the nonhomogeneous equation will be the general solution of the
corresponding homogeneous equation plus one or more additional terms that 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  t   by p  t   cy p  t  
                                                                             p

                             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  m 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, sinusoid, or polynomial, except with arbitrary coefficients that you
determine by substituting your guess into the nonhomogeneous equation.
ODE Lecture Notes                         Section 3.5                            Page 2 of 5

Practice:
  1. Solve the IVP y t   y t   2 y t   2e3t , y(0) = 1, y’(0) = 1.
ODE Lecture Notes                         Section 3.5                                 Page 3 of 5

  2. Solve the IVP y t   y t   2 y t   2sin t  , y(0) = 1, y’(0) = 1.
ODE Lecture Notes                         Section 3.5                                    Page 4 of 5

  3. Solve the IVP y t   y t   2 y t   2et sin t  , y(0) = 1, y’(0) = 1.
ODE Lecture Notes                         Section 3.5                            Page 5 of 5

  4. Solve the IVP y t   y t   2 y t   2et , y(0) = 1, y’(0) = 1.