Second Order Non-homogeneous Linear ODE (4)
Undetermined Coefficients Method (2)
Last lecture, we explore the solution to non-homogeneous second order ODE:
ay ''+ by '+ cy =
f (t )
(1)
 Kt m eγt

when f (t ) =  Kt m eγt cosηt ,
 Kt m eγt sin ηt

In this lecture, we would like to explore how to solve the non-homogeneous equation
when f(t) is the sum of such terms.
First we present
Superposition Principle: If y1 is a solution to
ay ' '+by '+cy = f (t )
and y 2 is a solution to
ay ' '+by '+cy = g (t )
Then y (t ) =c1 y1 +c 2 y 2 is a solution to
ay ' '+by '+cy =c1 f (t ) +c 2 g (t )
where c1 ,c 2 are constants.
You may easily prove this principle by definition.
a(c1 y1 + c2 y2 )' '+b(c1 y1 + c2 y2 )'+ c(c1 y1 + c2 y2 )
= c1 (ay1 ' '+by1 '+cy1 ) + c2 (ay2 ' '+by2 '+cy2 )
c1 f (t ) + c2 g (t )
By superposition principle, we have the following
Theorem: If y p is a particular solution to
ay ' '+by '+cy = f (t )
(1)
and c1 y1 + c 2 y2 is the general solution to the associated homogeneous equation:
ay ' '+by '+cy = 0 ,
then y (t ) =c1 y1 +c 2 y2 + y p is the general solution to non-homogeneous equation (1).
Example 1: a. Find a particular solution to y ' '− y '−2 y = 3e 2t
b. Find a particular solution to y ' '− y − 2 y = te t
c. Find a particular solution to y ' '− y − 2 y = te t + 3e 2t
d. Find the general solution to y ' '− y − 2 y = te t + 3e 2t
e. Solving IVP
 y ' '− y − 2 y = te t + 3e 2t

 y (0) = 1
 y ' ( 0) = 2

Method of Undetermined Coefficients
For ay ''+ by '+ cy =
Pm (t )eγ t , Pm (t ) is a polynomial with degree m, the particular solution
form
y=
t s ( Amt m +  + A1t + A0 )eγ t
p (t )
with
a. s=0 if γ is not a root of the associated characteristic equation ( ar 2 + br + c =
0 );
b. s=1 if γ is a simple root of the associated characteristic equation;
c. s=2 if γ is a double root of the associated characteristic equation.
For ay ''=
+ by '+ cy Pm (t )eγ t sin η t + Pn (t )eγ t cosη t , the particular solution form
y p=
(t ) t s ( Ak t k +  + A1t + A0 )eγ t cosη t
+ t s ( Bk t k +  + B1t + B0 )eγ t sin η t
where k=max(m, n), with
a. s=0 if is γ + iη not a root of the associated characteristic equation;
b. s=1 if γ + iη is a simple root of the associated characteristic equation;
Example 2: Write down the form of a particular solution to the equation
y ''− 2 y '− 8 y = (t 2 + 1)et cos 2t + (t 3 − t 2 )et sin 2t
Example 3: Find a particular solution to the equation
y ' '+4 y '−5 y = e −2t sin t + 2e −2t cos t