9 Method of undetermined coefficients

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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 .

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