Forced Second-Order IVP with Complex Roots: 04-BS-1, December 2019, Q1

Question 1: Forced Second-Order IVP with Complex Roots (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. $y''-12y'+45y=18\cos(3t)$, $y(0)=0$, $y'(0)=0$.

Find. The particular solution $y(t)$ satisfying the initial conditions.

Approach. Solve the homogeneous equation via the characteristic equation (complex roots), find a particular solution by undetermined coefficients, superpose, then fit the two initial conditions.

Homogeneous solution. $r^2-12r+45=0\Rightarrow r=\dfrac{12\pm\sqrt{144-180}}2=6\pm3i$, so $$y_h(t)=e^{6t}(C_1\cos3t+C_2\sin3t).$$ The real part $6\ne0$, so even though the forcing frequency $3$ equals the imaginary part of the roots, this is not resonance.
Particular solution. Try $y_p=A\cos3t+B\sin3t$. Substituting: $y_p''-12y_p'+45y_p=(36A-36B)\cos3t+(36A+36B)\sin3t$. Matching to $18\cos3t+0\sin3t$: $$36A-36B=18,\qquad 36A+36B=0\ \Longrightarrow\ A=\tfrac14,\ B=-\tfrac14.$$ $$y_p(t)=\tfrac14\cos3t-\tfrac14\sin3t.$$
General solution. $y(t)=e^{6t}(C_1\cos3t+C_2\sin3t)+\tfrac14\cos3t-\tfrac14\sin3t$.
Apply $y(0)=0$. $C_1+\tfrac14=0\Rightarrow C_1=-\tfrac14$.
Apply $y'(0)=0$. $y'(t)=6e^{6t}(C_1\cos3t+C_2\sin3t)+e^{6t}(-3C_1\sin3t+3C_2\cos3t)-\tfrac34\sin3t-\tfrac34\cos3t$. At $t=0$: $6C_1+3C_2-\tfrac34=0\Rightarrow6(-\tfrac14)+3C_2=\tfrac34\Rightarrow3C_2=\tfrac94\Rightarrow C_2=\tfrac34$.

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

Quantity	Result
Roots	$6\pm3i$
Particular solution	$\tfrac14\cos3t-\tfrac14\sin3t$
$C_1,\,C_2$	$-\tfrac14,\ \tfrac34$
$y(t)$	$e^{6t}(-\tfrac14\cos3t+\tfrac34\sin3t)+\tfrac14\cos3t-\tfrac14\sin3t$

Concept & Theory

Topic. Forced linear ODEs with complex characteristic roots; superposition of homogeneous and particular solutions under initial conditions.

Key relations. For $ay''+by'+cy=F\cos\omega t$ with complex roots $\alpha\pm i\beta$, $y_h=e^{\alpha t}(C_1\cos\beta t+C_2\sin\beta t)$; if $\omega\ne\beta$ or $\alpha\ne0$ the forcing is non-resonant and the undetermined-coefficients trial $A\cos\omega t+B\sin\omega t$ applies directly.

Why this works. Resonance in a damped/growing oscillator ($\alpha\ne0$) never truly occurs in the classical sense — the homogeneous solution's envelope $e^{\alpha t}$ already distinguishes it from any bounded forcing term, so the plain trial never collides with $y_h$ regardless of how close $\omega$ is to $\beta$.

Common pitfall. Assuming resonance merely because the forcing frequency $\omega=3$ numerically matches the imaginary part $\beta=3$ of the roots — the real part $\alpha=6\ne0$ means the homogeneous solution is not oscillatory-only, so no resonance trial is needed.

Source. Kreyszig, Advanced Engineering Mathematics, Ch. 2 (§2.7–2.8 Nonhomogeneous ODEs, Modeling Forced Oscillations).

Question 1 of 8: Forced Second-Order IVP with Complex Roots

Question 1: Forced Second-Order IVP with Complex Roots (20 marks)