Nonresonant sine and cosine forcing in a second order ODE

Expert intro: For oscillatory forcing, the correct particular solution usually mirrors the forcing frequency unless that frequency matches the homogeneous solution and causes resonance.

Detailed walkthrough

Set up the trial solution

For an oscillatory forcing term

$f(t)=a_n\cos(nt)+b_n\sin(nt),\qquad n\ne 3,$

a standard undetermined-coefficients guess for a second-order equation with natural frequency $3$ is

$y_p(t)=A\cos(nt)+B\sin(nt).$

Substitute this into the differential equation and solve for $A$ and $B$. In the common model

$y''+9y=f(t),$

the derivatives are

$y_p''(t)=-n^2A\cos(nt)-n^2B\sin(nt),$

so

$y_p''+9y_p=(9-n^2)A\cos(nt)+(9-n^2)B\sin(nt).$

Matching coefficients gives

$A=\frac{a_n}{9-n^2},\qquad B=\frac{b_n}{9-n^2}.$

Hence

$y_p(t)=\frac{a_n}{9-n^2}\cos(nt)+\frac{b_n}{9-n^2}\sin(nt),\qquad n\ne 3.$

What changes when n=3

If $n=3$, then the denominator $9-n^2$ becomes zero. That is not a harmless algebraic accident; it means the trial solution duplicates a homogeneous solution. The forcing frequency is resonant with the system, so the guess must be multiplied by $t$.

Why the naive analysis fails

The naive calculation assumes that the differential operator acts on $\cos(nt)$ and $\sin(nt)$ like a nonzero scalar multiple. That is true only when the forcing frequency is different from the natural frequency. At $n=3$, the operator sends the trial functions into the homogeneous solution space, so coefficient matching cannot determine $A$ and $B$ in the usual way.

Practical takeaway

For every nonresonant oscillatory term, the particular solution stays at the same frequency. For the resonant frequency, use a modified trial of the form

$t\big(A\cos(3t)+B\sin(3t)\big).$

💡 Pitfall guide

A frequent mistake is to plug in $n=3$ to the nonresonant formula and write down a division by zero without explaining the structural reason. The real issue is resonance: the forcing term lies in the same frequency space as the homogeneous solution. Another mistake is mixing up the roles of $a_n$ and $b_n$ after differentiation, which can happen if you do not keep the cosine and sine coefficients separate while matching terms.

🔄 Real-world variant

If the forcing term were $f(t)=4\cos(2t)-7\sin(2t)$, the same method would work with $n=2$, giving a nonresonant response proportional to $\cos(2t)$ and $\sin(2t)$. If instead the forcing were $f(t)=\cos(3t)$, the correct trial would be

$y_p(t)=t\big(A\cos(3t)+B\sin(3t)\big),$

because the ordinary guess collapses into the homogeneous solution. That resonance adjustment is the key difference between the two cases.

🔍 Related terms

resonant forcing, undetermined coefficients, harmonic oscillator