Mathematics for Physics · Chapter 23

Differential Equations

Physics equations are almost always differential equations — relations between a function and its derivatives. Knowing how to solve them is knowing how to solve physics.

Prerequisites

Calculus essentials (Ch. 22) \cdot Vectors (Ch. 21) \cdot Simple harmonic motion (Ch. 9) provides

physical motivation

Learning Goals

Solve first-order linear ODEs using the integrating factor method.
Solve second-order ODEs with constant coefficients by finding roots of the characteristic equation.
Classify damped oscillators as underdamped, overdamped, or critically damped from their parameters.
Find the steady-state amplitude of a driven oscillator and identify the resonance condition.
Use separation of variables to solve first-order separable ODEs and apply initial conditions.

23.1 What Is a Differential Equation?

A differential equation (DE) is an equation involving an unknown function and one or more of its derivatives. The order is the highest derivative that appears. An ordinary DE (ODE) has one independent variable; apartial DE (PDE) has multiple.

Every fundamental law of physics is a differential equation. Newton's second law: m d²x/dt² = F(x,t). Faraday's law: ∂B/∂t = −∇×E. Schrödinger's equation: iℏ ∂ψ/∂t = Ĥψ. Solving these equations — finding the function x(t) or ψ(x,t) consistent with initial conditions — is the central technical task of theoretical physics.

23.2 First-Order Linear ODEs

Definition 23.1 — First-Order Linear ODE

Standard form: dy/dx + P(x)y = Q(x). The general solution uses an

integrating factor

\mu(x) = exp(\intP dx

d/dx[\mu(x) y] = \mu(x) Q(x) \quad \to \quad y = (1/\mu) \int \muQ dx + C/\mu

The most important first-order ODE in physics is exponential growth/decay: dy/dt = ky. Its solution is y = y₀ eᵏᵗ. For k <0 (decay): radioactive decay (N = N₀ e^(−t/τ)), capacitor discharge (q = q₀ e^(−t/RC)), Newton's law of cooling (ΔT = ΔT₀ e^(−kt)).

Example 23.1 — RC Circuit — Capacitor Charging

$A capacitor (capacitance C) charges through a resistor R from a battery of EMF \varepsilon. The loop$ $equation gives \varepsilon = IR + q/C, with I = dq/dt. Solve for q(t$

Rewrite:

dq/dt + q/(RC) = \varepsilon/R \qquad (first-order linear ODE

Integrating factor:

\mu = e^(t/RC

Multiply through:

d/dt[q e^{t/RC}] = (\varepsilon/R) e^(t/RC

Integrate:

q e^{t/RC} = C\varepsilon e^{t/RC} + A \to q(t) = C\varepsilon + A e^(-t/RC

Initial condition:

q(0) = 0 \to A = -C\varepsilon \to q(t) = C\varepsilon(1 - e^(-t/RC

Result:

q(t) = C\varepsilon(1 - e^{-t/\tau}) where \tau = RC is the time constant. ✓

23.3 Second-Order Linear ODEs with Constant Coefficients

The equation m ẍ + b ẋ + kx = F(t) governs every oscillating system in physics: springs, LC circuits, pendulums, sound resonators. The general approach: try the ansatz x = e^(rt), substitute, and solve the characteristic equation.

Definition 23.2 — Characteristic Equation

For ay'' + by' + cy = 0, substitute y = e^(rt

ar^{2} + br + c = 0 \qquad (characteristic equation

Roots r_{1}, r_{2} determine the solution type

Real distinct (b^{2}

4ac): y = C_{1} e^{r_{1}t} + C_{2} e^{r_{2}t} \quad (overdamped

Complex (b^{2}

4ac): r = \alpha \pm i\omega \to y = e^{\alphat}(C_{1} cos \omegat + C_{2} sin \omegat) \quad (underdamped

Repeated (b^{2} = 4ac): y = (C_{1} + C_{2}t) e^{rt} \quad (critically damped

Example 23.2 — Simple Harmonic Oscillator

$Solve m ẍ + kx = 0 with initial conditions x(0) = A, ẋ(0) = 0$ .

Characteristic equation:

mr^{2} + k = 0 \to r^{2} = -k/m \to r = \pmi\sqrt(k/m) = \pmi\omega_{0}

General solution:

x(t) = C_{1} cos(\omega_{0}t) + C_{2} sin(\omega_{0}t), \qquad \omega_{0} = \sqrt(k/m

Apply x(0) = A:

C_{1} = A

Apply ẋ(0) = 0:ẋ =

-A\omega_{0} sin(\omega_{0}t) + C_{2}\omega_{0} cos(\omega_{0}t). At t=0: C_{2}\omega_{0} = 0 \to C_{2} = 0

Result:

x(t) = A cos(\omega_{0}t) — pure cosine, as expected for release from rest. ✓

Example 23.3 — Damped Oscillator

$Solve mẍ + bẋ + kx = 0 for the underdamped case (b^{2}$ < 4mk).

Characteristic equation:

mr^{2} + br + k = 0 \to r = [-b \pm \sqrt(b^{2}-4mk)] / 2m

Underdamped:

b^{2}

4mk \to \sqrt(b^{2}-4mk) is imaginary \to r = -\gamma \pm i\omega_d

Define:

\gamma = b/(2m) (decay rate), \qquad \omega_d = \sqrt(k/m - \gamma^{2}) = \sqrt(\omega_{0}^{2} - \gamma^{2}) (damped frequency

Solution:

x(t) = e^{-\gammat} (C_{1} cos \omega_d t + C_{2} sin \omega_d t) = A e^{-\gammat} cos(\omega_d t + \phi

Physics:

An exponentially decaying envelope \times oscillation. Amplitude halves every t_{1}/_{2} = ln2/\gamma

23.4 Particular Solutions and Resonance

For a driven oscillator mẍ + bẋ + kx = F₀ cos(ωt), the particular solution(steady-state response) has the form x_p = X cos(ωt − δ), where the amplitude is:

X = F_{0} / \sqrt[(k-m\omega^{2})^{2} + (b\omega)^{2}]

(23.1)

This is maximum when the driving frequency ω equals the natural frequency ω₀ = √(k/m) — this is resonance. At resonance (weak damping), the amplitude grows enormously: X_res = F₀/(bω₀). The quality factor Q = mω₀/b measures the sharpness of resonance — high Q means narrow, tall peak; low Q means broad, flat response.

Resonance is why bridges can be destroyed by rhythmic marching (Angers Bridge, 1850), why wine glasses shatter at their resonant frequency, and why MRI machines work (nuclear magnetic resonance). The mathematics is identical in all cases.

23.5 Separation of Variables

Many first-order ODEs can be solved by separating the variables:

Definition 23.3 — Separation of Variables

If dy/dx = f(x)g(y), rewrite as dy/g(y) = f(x)dx, then integrate both sides

\int dy/g(y) = \int f(x) dx + C

This directly yields y as a function of x (or implicitly). Works whenever the equation is separable — the right-hand side factors into a function of x times a function of y.

Example 23.4 — Radioactive Decay

$Carbon-14 decays with half-life t_{1}/_{2} = 5730 years. A bone sample contains 30% of the origi$ nal C-14. Find its age.

ODE:

dN/dt = -\lambdaN, where \lambda = ln2/t_{1}/_{2} = ln2/5730 yr^{-1}

Separate:

dN/N = -\lambda dt \to \intdN/N = -\lambda\intdt \to ln N = -\lambdat + const

Solution:

N(t) = N_{0} e^(-\lambdat

Apply N/N₀ = 0.30:

0.30 = e^{-\lambdat} \to -\lambdat = ln(0.30) = -1.204

Age:

t = 1.204/\lambda = 1.204 \times 5730/ln2 = 1.204 \times 8267 =

9,950 years

Definition 23.4 — Common Traps

General solutions need constants: initial or boundary conditions choose the physical solution.
Homogeneous and particular parts both matter: driven systems need the forced response.
Stability comes from eigenvalues: signs and real parts determine growth or decay.
Separation is not always valid: only separable equations allow variables to be split cleanly.

Exercises — 23.1–23.5 Differential Equations

A sample retains 30% of its original C-14 (half-life 5730 years). Find its age.

years

Straightforward

A mass–spring system has k = 400 N/m and m = 10 kg. Find the angular frequency \omega_{0}

rad/s

Straightforward

At t = RC (one time constant), what fraction of the maximum charge has a capacitor accumul

ated?

Intermediate

A spring-mass system has m=1 kg, k=4 N/m. Classify and solve for x(t) for: (a) b=0.5, (b

b=4, (c) b=10 N\cdots/m

Intermediate

Derive the steady-state amplitude X(\omega) for a driven oscillator mẍ+bẋ+kx = F_{0}cos(\omegat). At wh

at frequency is X maximum? Define the quality factor Q and explain its physical meaning.

Challenging

Key Takeaways

A differential equation relates a function to its derivatives; every physics law is a DE.
$Exponential growth/decay (dy/dt = ky \to y = y_{0}eᵏᵗ) governs radioactive decay, RC circuits$ , cooling.
$Characteristic equation (ar^{2}+br+c=0) classifies oscillators: underdamped (complex roots$ , overdamped (real distinct), critical (repeated).
$Simple harmonic oscillator: x(t) = A cos(\omega_{0}t+\phi), \omega_{0} = \sqrt(k/m$ .
$Damped oscillator: x(t) = Ae^{-\gammat}cos(\omega_d t+\phi) — exponentially decaying amplitude$ .
$Resonance at \omega \approx \omega_{0}: amplitude diverges as damping \to 0. Quality factor Q = \omega_{0}/\Delta\omega$ .