Questions: Complex Roots and Oscillatory Solutions

Question 1

A second-order linear ODE has characteristic roots r = -2 ± 3i. Which description best characterizes the general solution?

• A. A purely oscillatory solution with constant amplitude and frequency determined by the 3
• B. A damped oscillation whose amplitude shrinks exponentially over time due to the -2
• C. A growing oscillation whose amplitude increases exponentially due to the imaginary part 3i
• D. A non-oscillatory exponential decay, since the real part is negative

Answer: (B)

The roots α ± iβ = -2 ± 3i give the general solution y = e^(-2x)(c₁cos(3x) + c₂sin(3x)). The real part α = -2 controls the envelope: since α < 0, the factor e^(-2x) decays to zero, meaning the oscillation's amplitude shrinks — damped oscillation. The imaginary part β = 3 sets the oscillation frequency (angular frequency = 3). The solution oscillates (due to β ≠ 0) but with diminishing amplitude (due to α < 0).

Question 2

In the solution y = e^(αx)(c₁cos(βx) + c₂sin(βx)) from complex roots α ± iβ, which parameter controls how rapidly the oscillations cycle back and forth?

• A. α, because it appears in the exponential factor that wraps the entire expression
• B. β, because it is the argument of the cosine and sine functions
• C. c₁ and c₂, because they scale each oscillatory term independently
• D. The ratio α/β, because oscillation speed depends on both

Answer: (B)

β (the imaginary part of the roots) is the angular frequency of oscillation — it determines how many radians of oscillation occur per unit of x. Larger β means more cycles per unit x (faster oscillation). α (the real part) controls the amplitude envelope — whether the oscillation grows, decays, or remains constant. c₁ and c₂ are determined by initial conditions and set the specific trajectory, but don't change the frequency.

Question 3

If the characteristic equation of a second-order ODE yields complex roots r = α ± iβ with α < 0, the resulting solution will oscillate but with amplitude that decreases toward zero.

• T. True
• F. False

Answer: True

The general solution is y = e^(αx)(c₁cos(βx) + c₂sin(βx)). When α < 0, the factor e^(αx) → 0 as x → ∞, which 'envelopes' the oscillation and drives its amplitude to zero. This is damped oscillation — the physical model of a spring with friction or an electrical circuit with resistance. If α = 0, amplitude is constant (undamped); if α > 0, amplitude grows without bound (unstable).

Question 4

When the characteristic equation of a real-coefficient ODE has complex roots, the general solution of the ODE involves complex-valued (non-real) functions.

• T. True
• F. False

Answer: False

Even though the characteristic roots are complex, the general real solution is expressed entirely in real functions: y = e^(αx)(c₁cos(βx) + c₂sin(βx)). The key step is applying Euler's formula e^(iβx) = cos(βx) + i·sin(βx) and taking real and imaginary parts to obtain two real linearly independent solutions. This is a critical insight: complex roots do not imply complex solutions — the real and imaginary parts of the complex exponential solutions are themselves real-valued and span the solution space.

Question 5

Explain what the real part α and imaginary part β of complex characteristic roots r = α ± iβ each control in the general solution, and give a physical interpretation of each.

Model answer: α (the real part) controls the amplitude envelope of the oscillation. If α < 0, the amplitude decays exponentially (damped oscillation, like a pendulum with friction). If α = 0, amplitude stays constant (pure oscillation, like an ideal spring). If α > 0, amplitude grows exponentially (unstable oscillation). β (the imaginary part) controls the angular frequency of oscillation — how rapidly the solution cycles through its cosine-sine pattern. Larger β means more oscillations per unit of the independent variable.

The key insight is that α and β play completely independent roles — the real part governs 'how the amplitude changes over time' while the imaginary part governs 'how fast it oscillates.' This separation makes complex roots physically interpretable: the solution is a sinusoidal oscillation at frequency β whose amplitude is modulated exponentially at rate α.