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
   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 y h
( t ) is the solution to ay
  by
  cy

0 , and y p
( t ) is the solution to ay
  by
  cy
  
, then
   h
   y p
  is the most general solution to the nonhomogeneous equation because
          
 h
     h
   h
  

 ay
 p
   by
 p
   cy p
  

Theorem 3.5.1: The Difference of Nonhomogeneous Solutions Is the Homogeneous Solution
If Y
1
and Y
2
are two solutions of the nonhomogeneous equation
   y
          
, then their difference Y
1

Y
2 is a solution of the corresponding homogeneous solution
   y
        
0 . If, in addition, y
1
and y
2
are a fundamental set of solutions of the homogeneous equation, then
Y
1
 
2 c y t
1 1

2 2
 
, for certain constants c
1
and c
2
.
Theorem 3.5.2: General Solution of a Nonhomogeneous Equation
The general solution of the nonhomogeneous equation
   y
           be written in the form y
    
1 1
  
2 2
   
, can m where , y
1
and y
2
are a fundamental set of solutions of the corresponding homogeneous equation, c
1
and c
2
, 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
Practice:
1.
Solve the IVP
       
2
  
2 e
3 t , y(0) = 1, y’(0) = 1.
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ODE Lecture Notes Section 3.5
2.
Solve the IVP
       
2
  
2sin
, y(0) = 1, y’(0) = 1.
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ODE Lecture Notes Section 3.5
3.
Solve the IVP
       
2
   e
, y(0) = 1, y’(0) = 1.
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ODE Lecture Notes Section 3.5
4.
Solve the IVP
       
2
  
2 e
 t , y(0) = 1, y’(0) = 1.
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