Critically-Damped Forced Oscillator (Initial Value Problem): 04-BS-1, Undated paper, Q2

Question 2: Critically-Damped Forced Oscillator (Initial Value Problem) (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 critically damped, constant-coefficient second-order ODE with cosine forcing and initial conditions $y(0)=2$, $y'(0)=0$.

Find. $y(t)$.

Approach. Solve the repeated-root homogeneous equation, find a particular solution by undetermined coefficients, then fix the two constants from the initial conditions.

Homogeneous solution. $r^2+4r+4=(r+2)^2=0\Rightarrow r=-2$ (double root): $$y_h=(C_1+C_2t)e^{-2t}.$$
Particular solution. Try $y_p=A\cos2t+B\sin2t$. Substituting, $y_p''+4y_p'+4y_p=8B\cos2t-8A\sin2t$. Matching to $\cos2t$ gives $8B=1$, $-8A=0$, so $A=0,\ B=\tfrac18$: $$y_p=\tfrac18\sin2t.$$
General solution and initial conditions. $$y(t)=(C_1+C_2t)e^{-2t}+\tfrac18\sin2t.$$ $y(0)=C_1=2$. Differentiating, $y'(t)=C_2e^{-2t}-2(C_1+C_2t)e^{-2t}+\tfrac14\cos2t$, so $y'(0)=C_2-2C_1+\tfrac14=0\Rightarrow C_2=2(2)-\tfrac14=\tfrac{15}{4}$.

$$y(t)=\boxed{\left(2+\tfrac{15}{4}t\right)e^{-2t}+\tfrac18\sin2t}$$

Quantity	Result
Homogeneous roots	$r=-2$ (double)
$C_1,\ C_2$	$2,\ 15/4$
$y(t)$	$(2+\tfrac{15}{4}t)e^{-2t}+\tfrac18\sin2t$

Concept & Theory

Topic. Critically damped forced oscillator, undetermined coefficients, and initial-value fitting.

Key relations. A repeated root $r=-2$ gives homogeneous solutions $e^{-2t}$ and $t e^{-2t}$; a sinusoidal forcing at a frequency not matching the (real, repeated) homogeneous root needs no resonance modification.

Why this works. Because the forcing frequency ($\omega=2$) does not coincide with the (purely real) homogeneous exponent, the ordinary $A\cos2t+B\sin2t$ trial suffices; the two arbitrary constants are then pinned down uniquely by the two initial conditions.

Common pitfall. Forgetting the extra factor of $t$ in the second homogeneous solution for a repeated root (writing $C_1e^{-2t}+C_2e^{-2t}$, which is not linearly independent).

Source. Kreyszig, Advanced Engineering Mathematics, Ch. 2 (§2.4 Euler–Cauchy/repeated roots, §2.7–2.8 Undetermined Coefficients, Initial Value Problems).

Question 2 of 7: Critically-Damped Forced Oscillator (Initial Value Problem)

Question 2: Critically-Damped Forced Oscillator (Initial Value Problem) (20 marks)