9 Method of undetermined coefficients
Theorem 9.1 (Undetermined coefficients).
Consider the non-homogeneous ODE y ( n ) + a n − 1 y ( n − 1) + . . .
+ a
1 y 0 + a
0 y = f ( t ) .
If f ( t ) is a polynomial of degree m , then this ODE has a solution of the form y p
= t k ( b
0
+ b
1 t + . . .
+ b m t m ) .
If f ( t ) is e βt times a polynomial of degree m , then it has a solution of the form y p
= t k e βt ( b
0
+ b
1 t + . . .
+ b m t m ) .
If either f ( t ) = sin( βt ) or f ( t ) = cos( βt ), then it has a solution of the form y p
= t k
³
A sin( βt ) + B cos( βt )
´
.
In each of these cases, k ≥ 0 denotes the least integer such that y p which already appear in the solution y h does not contain terms to the associated homogeneous ODE.
Example 9.2.
Consider the non-homogeneous ODE y 00 − 4 y 0 + 4 y = t + e 2 t + cos(3 t ) .
In this case, the homogeneous solution y h is given by
λ 2 − 4 λ + 4 = 0 = ⇒ ( λ − 2) 2 = 0 = ⇒ y h
= c
1 e 2 t + c
2 te 2 t .
According to the theorem above, the non-homogeneous ODE has a solution of the form y p
= At + B + Ct 2 e 2 t + D sin(3 t ) + E cos(3 t ) .
Example 9.3.
Consider the non-homogeneous ODE y 00 − 3 y 0 + 2 y = t 2 + e 2 t + te 3 t .
In this case, the homogeneous solution y h is given by
λ 2 − 3 λ + 2 = 0 = ⇒ ( λ − 1)( λ − 2) = 0 = ⇒ y h
= c
1 e t + c
2 e 2 t .
According to the theorem above, the non-homogeneous ODE has a solution of the form y p
= At 2 + Bt + C + Dte 2 t + ( Et + F ) e 3 t .