Resonance
When the forcing term already appears in the complementary function, the trial particular integral diverges as ; for the resonant limit is .
Consider the ODE
which represents a simple harmonic oscillator driven at its natural frequency . We say that this oscillator is driven resonantly. We have the comcomplementary functions
Since is already in the complementary function, we consider detuning by looking at the slightly modified equation
with .
We try a particular integral of the form
We can see that must be zero since there is no term on the RHS. Substituting into the ODE, we get
Note that the limit does not exist since diverges. We can add in a complementary function to regularise the limit:
We know that this satisfies the detuned equation. Now, taking the limit , we have
Therefore, a particular integral for the resonant case is
General rule
The general rule: if the forcing term is a linear combination of linearly independent complementary functions, the particular integral is of the form
If the homogeneous equation itself is degenerate, higher powers may be needed:
for a 2nd order degenerate case.
For equidimensional ODEs the same idea applies with at resonance or : try , and if the homogeneous equation is degenerate, for a 2nd order degenerate case.
Related
Stated in
- Example 6.19 (Resonance)§6.5.1 Constant Coefficient ODEs