Complex Roots and Oscillations

In the previous chapter, we solved second-order homogeneous equations by finding roots of the characteristic equation. When those roots were real and distinct, we got exponential solutions. When they were real and repeated, we introduced a factor of $t$. But what happens when the characteristic equation has no real roots at all?

When the Discriminant Goes Negative

Consider the equation:

The characteristic equation is $r^2 + 1 = 0$, which gives $r^2 = -1$. There are no real solutions. But if we allow complex numbers, we get $r = \pm i$.

This is not a mathematical curiosity. This is where differential equations become beautiful. Complex roots do not mean the solution is imaginary. They reveal something profound: the solution oscillates.

Complex Conjugate Roots

Brief review of complex numbers: Complex numbers extend the real numbers by introducing $i$, defined by $i^2 = -1$. Every complex number has the form $a + bi$ where $a$ and $b$ are real. The conjugate of $a + bi$ is $a - bi$. Complex numbers can be added, multiplied, and exponentiated following the usual rules of algebra, with $i^2$ replaced by $-1$ whenever it appears.

When the characteristic equation $ar^2 + br + c = 0$ has a negative discriminant, the quadratic formula gives:

We write these roots as:

where $\alpha = -\frac{b}{2a}$ is the real part and $\beta = \frac{\sqrt{4ac - b^2}}{2a}$ is the imaginary part.

Notice that complex roots always come in conjugate pairs: if $\alpha + \beta i$ is a root, so is $\alpha - \beta i$. The real part $\alpha$ determines whether solutions grow or decay. The imaginary part $\beta$ determines the frequency of oscillation.

Euler's Formula: The Bridge

How do we turn complex exponentials into real solutions? The key is Euler's remarkable formula:

This identity connects the exponential function to trigonometry. Where does it come from? One way to see it is through Taylor series: expanding both sides as power series and verifying they match. Another perspective is to define $e^{i\theta}$ to be $\cos\theta + i\sin\theta$, which makes complex exponentials consistent with all the usual rules of calculus.

The formula tells us that complex exponentials naturally produce sines and cosines. Note also that $e^{a+bi} = e^a \cdot e^{bi}$, so we can always separate a complex exponential into a real exponential times a unit complex exponential.

For our characteristic roots $r = \alpha \pm \beta i$, the complex solutions would be:

and

Extracting Real Solutions

We need real solutions for a real differential equation. The trick is to take linear combinations of the complex solutions that eliminate the imaginary parts.

Adding the two complex solutions:

Subtracting them:

Dividing by the appropriate constants gives us two real, linearly independent solutions:

The general solution is therefore:

This is the form you should commit to memory. When you see complex conjugate roots $\alpha \pm \beta i$, the solution involves:

• An exponential factor $e^{\alpha t}$ controlling growth or decay
• Oscillatory factors $\cos\beta t$ and $\sin\beta t$ controlling the frequency

The Physical Meaning

The structure of this solution has direct physical interpretation.

The real part $\alpha$ controls amplitude behavior:

• When $\alpha < 0$: the amplitude decays exponentially. Energy is being dissipated.
• When $\alpha = 0$: the amplitude stays constant. Pure, sustained oscillation.
• When $\alpha > 0$: the amplitude grows exponentially. Energy is being pumped in.

The imaginary part $\beta$ controls frequency:

• Larger $\beta$ means faster oscillation
• The period of oscillation is $T = \frac{2\pi}{\beta}$

In physical systems, $\alpha < 0$ corresponds to damping (friction, air resistance, electrical resistance). Pure oscillation with $\alpha = 0$ is an idealization that never quite happens in reality. Growing oscillations with $\alpha > 0$ require an external energy source and typically lead to instability.

Amplitude-Phase Form

The general solution $y = e^{\alpha t}(C_1\cos\beta t + C_2\sin\beta t)$ can be written in an equivalent form that is sometimes more useful:

where $A$ is the amplitude and $\phi$ is the phase shift.

The conversion formulas are:

Note on the phase: The formula $\phi = \arctan(C_2/C_1)$ works when $C_1 > 0$. For the general case, use $\cos\phi = C_1/A$ and $\sin\phi = C_2/A$ to determine $\phi$ in the correct quadrant.

The amplitude-phase form makes it easy to read off physical quantities directly. The amplitude $A$ tells you how big the oscillations are. The phase $\phi$ tells you when the oscillation reaches its maximum. The damping factor $e^{\alpha t}$ modulates the overall size.

Damping: Three Regimes

The discriminant $b^2 - 4ac$ of the characteristic equation determines the nature of the roots and hence the behavior of solutions. For the equation $ay'' + by' + cy = 0$:

Underdamped ($b^2 < 4ac$): Complex roots. The solution oscillates while decaying. This is what we have been studying in this chapter.

Critically Damped ($b^2 = 4ac$): A repeated real root. The solution decays as fast as possible without oscillating. This is the boundary case.

Overdamped ($b^2 > 4ac$): Two distinct real roots. The solution decays without oscillation, but more slowly than critical damping.

Critical damping is often the design goal in engineering. A car's shock absorbers, for instance, should be critically damped: you want the car to return to equilibrium as quickly as possible after hitting a bump, without bouncing up and down (underdamped) or taking too long to settle (overdamped).

A Physical Example: The Damped Spring

A mass attached to a spring with friction satisfies:

where $m$ is mass, $b$ is the damping coefficient (friction), and $k$ is the spring constant.

The characteristic equation is $mr^2 + br + k = 0$, giving roots:

When friction is small ($b^2 < 4mk$), we get complex roots. The mass oscillates back and forth, but each swing is a little smaller than the last as energy is lost to friction.

The Pure Oscillator

Setting the damping to zero ($b = 0$) gives the equation:

The roots are $r = \pm i\sqrt{k/m} = \pm i\omega_0$, where $\omega_0 = \sqrt{k/m}$ is called the natural frequency.

The solution is:

This is simple harmonic motion: pure oscillation that continues forever at a fixed amplitude. It is an idealization, since real systems always have some friction. But it is the starting point for understanding all oscillatory behavior.

The natural frequency $\omega_0$ depends only on the physical parameters of the system: stiffer springs (larger $k$) and lighter masses (smaller $m$) oscillate faster.

Solving Initial Value Problems

Given an equation with complex roots and initial conditions, here is the procedure:

1. Write the characteristic equation and find the roots $r = \alpha \pm \beta i$
2. Write the general solution: $y = e^{\alpha t}(C_1\cos\beta t + C_2\sin\beta t)$
3. Apply $y(0) = y_0$ to find one equation for $C_1$ and $C_2$
4. Differentiate: $y' = \alpha e^{\alpha t}(C_1\cos\beta t + C_2\sin\beta t) + e^{\alpha t}(-\beta C_1\sin\beta t + \beta C_2\cos\beta t)$
5. Apply $y'(0) = v_0$ to find the second equation
6. Solve for $C_1$ and $C_2$

Example: Solve $y'' + 2y' + 5y = 0$ with $y(0) = 1$, $y'(0) = 0$.

The characteristic equation $r^2 + 2r + 5 = 0$ has roots $r = -1 \pm 2i$.

So $\alpha = -1$ and $\beta = 2$. The general solution is:

From $y(0) = 1$: $C_1 = 1$.

Differentiating: $y' = -e^{-t}(C_1\cos 2t + C_2\sin 2t) + e^{-t}(-2C_1\sin 2t + 2C_2\cos 2t)$

From $y'(0) = 0$: $-C_1 + 2C_2 = 0$, so $C_2 = 1/2$.

The solution is:

Or in amplitude-phase form: $y = \frac{\sqrt{5}}{2}e^{-t}\cos(2t - \arctan(1/2))$.

Key Takeaways

• Complex roots $r = \alpha \pm \beta i$ produce oscillating solutions
• The general solution is $y = e^{\alpha t}(C_1\cos\beta t + C_2\sin\beta t)$
• The real part $\alpha$ controls decay (negative) or growth (positive)
• The imaginary part $\beta$ controls the oscillation frequency
• Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ is the bridge between complex exponentials and trigonometry
• Underdamped systems oscillate while decaying; critically damped systems decay fastest without oscillating; overdamped systems decay slowly without oscillating
• The amplitude-phase form $y = Ae^{\alpha t}\cos(\beta t - \phi)$ is equivalent and often more physically intuitive