Second-Order IVP by Undetermined Coefficients: 04-BS-1, May 2013, Q1

Question 1: Second-Order IVP by Undetermined Coefficients (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. Linear, constant-coefficient, nonhomogeneous ODE $y''-4y'+8y=5\cos(2t)$ with initial conditions $y(0)=0,\ y'(0)=0$.

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

Approach. Solve the homogeneous equation via the characteristic equation, find a particular solution by undetermined coefficients (no resonance, since the forcing frequency does not match the homogeneous solution's oscillation frequency once the $e^{2t}$ envelope is accounted for), then fix the two constants from the initial conditions.

Solve the homogeneous equation. The characteristic equation is $$r^2-4r+8=0 \quad\Rightarrow\quad r=\frac{4\pm\sqrt{16-32}}{2}=2\pm 2i.$$ So $y_h(t)=e^{2t}\left(C_1\cos 2t + C_2\sin 2t\right)$.
Find a particular solution. Since $\cos(2t)$ is not itself a solution of the homogeneous equation (the homogeneous modes carry the $e^{2t}$ envelope), try $y_p=A\cos 2t + B\sin 2t$. Substituting: $$y_p''-4y_p'+8y_p = (4A-8B)\cos 2t + (8A+4B)\sin 2t.$$ Matching to $5\cos 2t+0\sin2t$ gives $4A-8B=5$ and $8A+4B=0$. The second gives $B=-2A$; substituting into the first, $4A+16A=5\Rightarrow A=0.25,\ B=-0.5$. So $$y_p(t)=0.25\cos 2t-0.5\sin 2t.$$
Assemble the general solution. $$y(t)=e^{2t}\left(C_1\cos 2t+C_2\sin 2t\right)+0.25\cos 2t-0.5\sin 2t.$$
Apply the initial conditions. $y(0)=C_1+0.25=0\Rightarrow C_1=-0.25$. Differentiating and evaluating at $t=0$: $$y'(0)=2C_1+2C_2-1=0 \;\Rightarrow\; 2(-0.25)+2C_2-1=0 \;\Rightarrow\; C_2=0.75.$$

$$y(t)=\boxed{e^{2t}\left(-0.25\cos 2t+0.75\sin 2t\right)+0.25\cos 2t-0.5\sin 2t}$$

Quantity	Result
$y(t)$	$e^{2t}(-0.25\cos 2t+0.75\sin 2t)+0.25\cos 2t-0.5\sin 2t$

Concept & Theory

Topic. Linear constant-coefficient ODEs — undetermined coefficients with complex characteristic roots.

Key relations. Characteristic equation $r^2+pr+q=0$; complex-conjugate roots $\alpha\pm i\beta$ give $y_h=e^{\alpha t}(C_1\cos\beta t+C_2\sin\beta t)$; undetermined-coefficients trial for $\cos\omega t$ forcing is $A\cos\omega t+B\sin\omega t$ unless that trial solves the homogeneous equation (resonance), in which case multiply by $t$.

Why this works. The homogeneous solution here oscillates at frequency 2 but is damped/grown by $e^{2t}$ — it is not a pure $\cos 2t$/$\sin 2t$ term, so the plain trial $A\cos 2t+B\sin 2t$ is linearly independent of $y_h$ and works without a resonance multiplier. The two initial conditions then pin down $C_1,C_2$ uniquely because the general solution has exactly two free constants for a second-order linear ODE.

Common pitfall. Forgetting the envelope check and assuming resonance is needed whenever the forcing frequency numerically matches the imaginary part of the roots — resonance only applies when the *undamped* forcing term itself solves the homogeneous equation (i.e. $\alpha=0$). Also, a sign slip differentiating the $e^{2t}$ product (product rule on two factors) is the most common arithmetic error.

Source. Kreyszig, Advanced Engineering Mathematics, Ch. 2 (Second-Order Linear ODEs, §2.7 Undetermined Coefficients).

Question 1 of 8: Second-Order IVP by Undetermined Coefficients

Question 1: Second-Order IVP by Undetermined Coefficients (20 marks)