Second-Order ODE with Complex Conjugate Roots

Setup

The ODE $y'' + 4 y = 0$ is the harmonic oscillator with angular frequency $\omega = 2$. Springs, pendulums (small-amplitude), LC circuits, and sound-wave equations all boil down to this shape.

Step-by-step solution

Step 1 — Characteristic equation. $r^{2} + 4 = 0 \implies r^{2} = -4 \implies r = \pm 2 i$

Discriminant $\Delta = 0 - 16 = -16 < 0$ → complex conjugate roots.

Step 2 — Build the real general solution.

With roots $r = \alpha \pm i \beta$ the general real-valued solution is $y(x) = e^{\alpha x} \bigl(C_1 \cos \beta x + C_2 \sin \beta x\bigr).$

Here $\alpha = 0$, $\beta = 2$: $\boxed{\, y(x) = C_1 \cos(2 x) + C_2 \sin(2 x) \,}$

Why cos and sin, not $e^{\pm i \beta x}$?

Technically the general complex solution is $y = A e^{i \beta x} + B e^{-i \beta x}$. Using Euler's formula $e^{i\theta} = \cos\theta + i \sin\theta$: $A e^{i\beta x} + B e^{-i\beta x} = (A + B) \cos(\beta x) + i (A - B) \sin(\beta x)$

For a real solution we need $(A + B) \in \mathbb{R}$ and $i(A - B) \in \mathbb{R}$, which forces $B = \bar{A}$. Writing $A = \frac{1}{2}(C_1 - i C_2)$ recovers $y = C_1 \cos \beta x + C_2 \sin \beta x$ with $C_1, C_2 \in \mathbb{R}$.

That's the bridge from the complex algebra of the characteristic equation to the real-valued physical solution.

Verification

$y = C_1 \cos 2x + C_2 \sin 2x$ $y' = -2 C_1 \sin 2x + 2 C_2 \cos 2x$ $y'' = -4 C_1 \cos 2x - 4 C_2 \sin 2x = -4 y$

So $y'' + 4 y = -4 y + 4 y = 0$. ✓

Amplitude–phase form

The same solution can be written as a single cosine with amplitude $A$ and phase $\varphi$: $y(x) = A \cos(\beta x - \varphi)$ where $A = \sqrt{C_1^{2} + C_2^{2}}, \qquad \tan \varphi = \frac{C_2}{C_1}.$

Expanding $\cos(\beta x - \varphi) = \cos(\beta x) \cos\varphi + \sin(\beta x) \sin\varphi$ gives $C_1 = A \cos\varphi$, $C_2 = A \sin\varphi$ — matching.

When to prefer which form?

• $C_1, C_2$ form is great for applying initial conditions cleanly (each IC plugs straight into one constant).
• Amplitude–phase form is great for reading off size of oscillation and timing relative to a reference.

Initial value problem

$y(0) = 3$, $y'(0) = -4$. $y(0) = C_1 = 3$ $y'(0) = 2 C_2 = -4 \implies C_2 = -2$ $y(x) = 3 \cos 2x - 2 \sin 2x$

Amplitude $A = \sqrt{9 + 4} = \sqrt{13}$; phase $\varphi = \arctan(-2/3) \approx -0.588$ rad.

Physical interpretation — simple harmonic motion

$m y'' + k y = 0$ describes a mass on a spring with no friction. Characteristic roots $r = \pm i \sqrt{k/m}$, so solutions are $\cos(\omega t)$, $\sin(\omega t)$ with natural frequency $\omega = \sqrt{k/m}.$

Our problem, $y'' + 4 y = 0$, has $k/m = 4$, $\omega = 2$ — the oscillation completes one full cycle every $T = 2\pi / \omega = \pi$ units of $x$.

With damping — $\alpha \ne 0$

If instead the ODE has a first-derivative term, $y'' + 2 \gamma y' + \omega_0^2 y = 0$, the roots are $r = -\gamma \pm \sqrt{\gamma^{2} - \omega_0^{2}}.$

• $\gamma < \omega_0$ (under-damped): complex roots, $\alpha = -\gamma < 0$, $\beta = \sqrt{\omega_0^2 - \gamma^2}$. Solution $y = e^{-\gamma x} (C_1 \cos \beta x + C_2 \sin \beta x)$ is an oscillation inside a decaying envelope.
• $\gamma = \omega_0$ (critically damped): repeated real root, see #182/#185.
• $\gamma > \omega_0$ (over-damped): real distinct roots, #183. No oscillation.

Our pure-oscillator $y'' + 4 y = 0$ is the un-damped limit ($\gamma = 0$, $\alpha = 0$).

Common mistakes

• Writing $y = C_1 e^{2 i x} + C_2 e^{-2 i x}$ as a final answer for a real problem. Complex exponentials are a useful intermediate, but the answer to a real ODE with real initial conditions should be real-valued.
• Confusing frequency $\omega$ with period $T$. $T = 2\pi / \omega$.
• Using $\beta = |r|$ with complex $r$. $\beta$ is the imaginary part, not the magnitude.
• Dropping the $e^{\alpha x}$ envelope for damped oscillations. Pure oscillation only occurs when $\alpha = 0$ (roots on the imaginary axis).

Try it in the visualization

Slide $\omega$ (the coefficient of $y$ in the ODE) and watch the period shrink or grow. Enable damping to slide the roots off the imaginary axis — see the oscillation decay or explode. Overlay the amplitude–phase envelope to see where the peaks and zero-crossings land.