LECTURE 19

Differential Equations

Laws of change — equations that describe how systems evolve

Uranium-238 decays to thorium-234, which decays to protactinium-234:

$^{238}\text{U} \xrightarrow{k_1} {}^{234}\text{Th} \xrightarrow{k_2} {}^{234}\text{Pa}$

The amount of thorium depends on:

  • How fast uranium produces it (+k₁[U])
  • How fast thorium decays (-k₂[Th])

$$\frac{d[\text{Th}]}{dt} = k_1[\text{U}] - k_2[\text{Th}]$$

This is a differential equation — an equation relating a function to its derivatives.

Solving it tells us how [Th] changes over time.

Recognition

Differential equations combine our primitives:
CHANGE (dy/dt) + RATE (how fast) + ACCUMULATION (integration to solve)

A DE is a statement about rates. Solving it reveals the function.

Classification

Property Description
Order Highest derivative (1st, 2nd, etc.)
Linearity y and derivatives to first power only
ODE vs PDE One vs multiple independent variables

First-Order Kinetics

The Fundamental Equation

$$\frac{d[A]}{dt} = -k[A]$$

Solution: $[A] = [A]_0 e^{-kt}$

Interactive: First-Order Decay
Half-life t₁/₂ = ln(2)/k: 6.93 s
Key property: Half-life is constant!

Solution Method (Separation of Variables)

$$\frac{d[A]}{[A]} = -k \, dt$$

$$\int \frac{d[A]}{[A]} = -k \int dt$$

$$\ln[A] = -kt + C \quad \Rightarrow \quad [A] = [A]_0 e^{-kt}$$

Consecutive Reactions: A → B → C

B is produced from A and consumed to form C:

$$\frac{d[B]}{dt} = k_1[A] - k_2[B]$$

Interactive: Consecutive Reactions
[B]ₘₐₓ at t = 13.86 s
Maximum [B]: 0.500 M

Solutions

$[A] = [A]_0 e^{-k_1 t}$

$[B] = \frac{k_1[A]_0}{k_2 - k_1}\left(e^{-k_1 t} - e^{-k_2 t}\right)$

$[C] = [A]_0 - [A] - [B]$

Second-Order Linear ODEs

Standard Form (Constant Coefficients)

$$ay'' + by' + cy = 0$$

Method: Try $y = e^{rx}$ → Characteristic equation: $ar^2 + br + c = 0$

Three Cases

Case 1: Two Distinct Real Roots r₁, r₂

$$y = C_1 e^{r_1 x} + C_2 e^{r_2 x}$$

Case 2: Repeated Root r

$$y = (C_1 + C_2 x)e^{rx}$$

Case 3: Complex Roots α ± βi

$$y = e^{\alpha x}(C_1 \cos\beta x + C_2 \sin\beta x)$$

This gives oscillatory solutions!

Damped Harmonic Oscillator

$$m\frac{d^2x}{dt^2} + \gamma\frac{dx}{dt} + kx = 0$$

Divide by m: $x'' + 2\beta x' + \omega_0^2 x = 0$

Interactive: Damped Oscillator
Regime: Underdamped
β² - ω₀² = -8.75
Condition Regime Behavior
β < ω₀ Underdamped Oscillates with decay
β = ω₀ Critical Fastest decay, no oscillation
β > ω₀ Overdamped Slow exponential decay

Quantum: Particle in a Box

Time-independent Schrödinger equation with V = 0 inside:

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$$

Interactive: Wavefunctions and Probability
Energy Eₙ ∝ n² = 1
Internal nodes: 0

Solutions

$$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$$

$$E_n = \frac{n^2 h^2}{8mL^2}$$

Boundary conditions + DE → quantized energy!

Numerical Methods: Euler's Method

When analytical solutions fail, step forward numerically:

$$y_{n+1} = y_n + h \cdot f(x_n, y_n)$$

Interactive: Euler vs Exact for dy/dx = -y
Exact y(3) = e⁻³: 0.0498
Euler y(3): 0.0574
Relative error: 15.3%

For serious work: Use Runge-Kutta (RK4) or adaptive methods (Julia's DifferentialEquations.jl, Python's scipy.integrate).

Summary: DE Toolkit

First-Order

Type Form Method
Separable dy/dx = g(x)h(y) Separate, integrate
Linear dy/dx + Py = Q Integrating factor μ = e^(∫Pdx)

Second-Order (Constant Coefficients)

Roots Solution Form
r₁, r₂ distinct real C₁e^(r₁x) + C₂e^(r₂x)
r repeated (C₁ + C₂x)e^(rx)
α ± βi complex e^(αx)(C₁cos βx + C₂sin βx)

Chemistry Applications

System Equation Solution
1st-order kinetics d[A]/dt = -k[A] [A]₀e^(-kt)
2nd-order kinetics d[A]/dt = -k[A]² [A]₀/(1+kt[A]₀)
Harmonic oscillator x'' + ω²x = 0 A cos(ωt + φ)