Forced Second-Order IVP: 04-BS-1, May 2016, Q2

Question 2: Forced Second-Order IVP 20 marks

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A constant-coefficient linear ODE with complex characteristic roots, forced at $\cos(3t)$, and zero initial conditions.

Find. $y(t)$ satisfying the IVP.

Approach. Solve the homogeneous equation (complex roots), find a particular solution by undetermined coefficients (no resonance, since the homogeneous modes carry an exponentially growing envelope rather than a bounded oscillation at frequency 3), then fix the two constants from the initial conditions.

Homogeneous solution. $r^2-12r+45=0\Rightarrow r=\dfrac{12\pm\sqrt{144-180}}{2}=6\pm3i$, so $$y_h(t)=e^{6t}(C_1\cos3t+C_2\sin3t).$$
Particular solution. Try $y_p=A\cos3t+B\sin3t$. Then $y_p'=-3A\sin3t+3B\cos3t$ and $y_p''=-9A\cos3t-9B\sin3t$. Substituting into $y_p''-12y_p'+45y_p$ and collecting $\cos3t,\sin3t$ terms: $$(36A-36B)\cos3t+(36A+36B)\sin3t=18\cos3t.$$ Matching coefficients: $36A-36B=18$ and $36A+36B=0$. The second gives $B=-A$; substituting into the first, $72A=18\Rightarrow A=\tfrac14,\ B=-\tfrac14$. So $y_p=\tfrac14\cos3t-\tfrac14\sin3t$.
General solution and first initial condition. $$y(t)=e^{6t}(C_1\cos3t+C_2\sin3t)+\tfrac14\cos3t-\tfrac14\sin3t.$$ $y(0)=C_1+\tfrac14=0\Rightarrow C_1=-\tfrac14$.
Second initial condition. Differentiating (product rule on the $e^{6t}$ term), $$y'(t)=e^{6t}\big[(6C_1+3C_2)\cos3t+(6C_2-3C_1)\sin3t\big]-\tfrac34\sin3t-\tfrac34\cos3t,$$ so $y'(0)=6C_1+3C_2-\tfrac34=0\Rightarrow 3C_2=\tfrac34-6(-\tfrac14)=\tfrac34+\tfrac32=\tfrac94\Rightarrow C_2=\tfrac34$.

$$y(t)=\boxed{e^{6t}\left(-\tfrac14\cos3t+\tfrac34\sin3t\right)+\tfrac14\cos3t-\tfrac14\sin3t}$$

Quantity	Result
Homogeneous roots	$r=6\pm3i$
Particular solution	$y_p=\tfrac14\cos3t-\tfrac14\sin3t$
$y(t)$	$e^{6t}\left(-\tfrac14\cos3t+\tfrac34\sin3t\right)+\tfrac14\cos3t-\tfrac14\sin3t$

Concept & Theory

Topic. Forced constant-coefficient linear ODE with complex characteristic roots; undetermined coefficients; initial-value problems.

Key relations. For $ay''+by'+cy=F_0\cos\omega t$ with complex roots $\alpha\pm i\beta$ and $\beta\ne\omega$ (no resonance), the steady-state particular solution has the same frequency as the forcing: $y_p=A\cos\omega t+B\sin\omega t$, found by direct substitution and matching coefficients.

Why this works. Here $\alpha=6\ne0$, so the homogeneous solution $e^{6t}(C_1\cos3t+C_2\sin3t)$ grows exponentially rather than oscillating in a bounded way at frequency 3 — it is linearly independent of the bounded functions $\cos3t,\sin3t$. This linear independence is precisely what guarantees the un-multiplied trial $A\cos3t+B\sin3t$ works directly, with no need to multiply by $t$ as one would in a resonant case where the forcing frequency matched the homogeneous oscillation frequency.

Common pitfall. Arithmetic slips solving the $2\times2$ system for $A,B$ — in particular, mixing up which linear combination of $A,B$ came from matching $\cos3t$ coefficients versus $\sin3t$ coefficients. Also, dropping the cross term $6C_2-3C_1$ (or $6C_1+3C_2$) when differentiating $e^{6t}(C_1\cos3t+C_2\sin3t)$ via the product rule — each of the two terms inside the parentheses differentiates to two pieces (one from the exponential envelope, one from the trig factor), and it is easy to keep only one piece by mistake.

Source. Kreyszig, Advanced Engineering Mathematics, Ch. 2 (§2.8 Modeling: Forced Oscillations, Resonance).

Question 2 of 8: Forced Second-Order IVP

Question 2: Forced Second-Order IVP 20 marks