# Differential Equations

Canonical URL: https://shipslides.com/d/mathematics-differential-equations
Raw viewer URL: https://content.shipslides.com/d/mathematics-differential-equations/raw
Category: Mathematics
Slides: 31
Updated: 2026-05-17T20:51:32.834Z
Tags: mathematics, differential, equations

## Summary

The Language of Change and Motion Key sections include: Differential Equations; What Is a Differential Equation?; Classification of ODEs; First-Order ODEs: Separable & Linear; Existence and Uniqueness; Second-Order Linear ODEs; The Harmonic Oscillator; Systems of ODEs; Phase Plane Analysis; Laplace Transform.

## Slide Outline

1. Differential Equations
2. What Is a Differential Equation?
3. Classification of ODEs
4. First-Order ODEs: Separable & Linear
5. Existence and Uniqueness
6. Second-Order Linear ODEs
7. The Harmonic Oscillator
8. Systems of ODEs
9. Phase Plane Analysis
10. Laplace Transform
11. Series Solutions and Special Functions
12. Sturm-Liouville Theory
13. Partial Differential Equations
14. Separation of Variables
15. Fourier Series and Transforms
16. The Navier-Stokes Equations
17. Nonlinear Dynamics and Chaos
18. Bifurcation Theory
19. Numerical Methods for ODEs
20. Stiff Systems
21. Numerical Methods for PDEs
22. Maxwell's Equations
23. The Schrodinger Equation
24. Einstein's Field Equations
25. Biological Applications
26. Financial Mathematics
27. Variational Methods
28. Historical Timeline
29. Modern Frontiers
30. Software Ecosystem
31. Key Takeaways

## Slide Transcript

### Slide 1: Differential Equations

- The Language of Change and Motion
- Differential equations relate functions to their derivatives -- encoding how systems evolve through time, space, or any continuous parameter. From Newton's second law to the Black-Scholes equation, they are the mathematical backbone of physics, engineering, biology, and finance.
- This deck covers ordinary and partial differential equations: theory, solution methods, and applications across the sciences.

### Slide 2: What Is a Differential Equation?

- An equation involving an unknown function and one or more of its derivatives. The goal: find the function(s) satisfying the equation.
- Ordinary (ODE)
- One independent variable. Example: dy/dx = ky models exponential growth. Newton's law: m*x'' = F(x,x',t).
- dy/dt = f(t, y)
- Partial (PDE)
- Multiple independent variables and partial derivatives. Example: heat equation, wave equation, Navier-Stokes.
- du/dt = k * d^2u/dx^2

### Slide 3: Classification of ODEs

- PropertyDescriptionExample
- OrderHighest derivative presenty'' + y = 0 is 2nd order
- Lineary and derivatives appear linearlyy'' + p(t)y' + q(t)y = g(t)
- NonlinearProducts/powers of y or derivativesy' = y^2 (Riccati)
- Autonomoust does not appear explicitlyy' = y(1-y) (logistic)
- HomogeneousRight-hand side = 0y'' + y = 0
- Constant coefficientsCoefficients don't depend on ty'' + 3y' + 2y = 0

### Slide 4: First-Order ODEs: Separable & Linear

- Separable Equations
- Form: dy/dx = f(x)g(y). Separate variables and integrate both sides.
- dy/g(y) = f(x) dx
- Example: dy/dx = xy gives y = Ce^(x^2/2). Works whenever the equation factors into a product of functions of x and y alone.
- First-Order Linear
- Form: y' + P(x)y = Q(x). Solved by integrating factor mu(x) = exp(integral P dx).
- y = (1/mu) * integral[mu * Q dx]
- Example: y' + 2y = e^(-x) gives y = e^(-x) + Ce^(-2x). Guaranteed existence and uniqueness when P, Q are continuous.

### Slide 5: Existence and Uniqueness

- When does a solution exist? When is it unique? These foundational questions were settled in the 19th century.
- Picard-Lindelof Theorem
- If f(t,y) is continuous and Lipschitz in y, then y' = f(t,y) with y(t_0) = y_0 has a unique local solution. The Lipschitz condition prevents "branching."
- Peano's Theorem
- If f is merely continuous (no Lipschitz), existence is guaranteed but uniqueness may fail. Example: y' = y^(2/3), y(0)=0 has infinitely many solutions.
- Blow-up
- Solutions may not exist globally. y' = y^2, y(0)=1 gives y = 1/(1-t), which explodes at t=1. Finite-time singularities are physically meaningful.

### Slide 6: Second-Order Linear ODEs

- The most important class: y'' + p(t)y' + q(t)y = g(t). Ubiquitous in mechanics, circuits, and wave phenomena.
- Homogeneous (g=0)
- Solution space is 2-dimensional (superposition principle)
- General solution: y = c1*y1 + c2*y2 (linearly independent)
- Wronskian W(y1,y2) != 0 certifies independence
- Constant coefficients: characteristic equation r^2 + pr + q = 0
- Characteristic Roots
- Distinct real roots r1, r2: y = c1*e^(r1*t) + c2*e^(r2*t)
- Repeated root r: y = (c1 + c2*t)*e^(r*t)
- Complex roots a +/- bi: y = e^(at)*(c1*cos(bt) + c2*sin(bt))
- Oscillatory behavior from complex roots (springs, circuits)

### Slide 7: The Harmonic Oscillator

- The single most important differential equation in physics: mx'' + bx' + kx = F(t). Models springs, pendulums, circuits, molecular vibrations.
- Undamped (b=0)
- x'' + omega^2 * x = 0. Solution: x = A*cos(omega*t + phi). Pure oscillation at natural frequency omega = sqrt(k/m). Energy conserved forever.
- Underdamped (b^2 Oscillates with exponentially decaying amplitude. Envelope: e^(-bt/2m). Frequency slightly less than natural. Most physical oscillators.
- Overdamped (b^2 > 4mk)
- No oscillation -- exponential decay with two time constants. Door closers, shock absorbers. Returns to equilibrium without overshoot.
- Critically Damped (b^2 = 4mk)
- Fastest return to equilibrium without oscillation. (c1 + c2*t)*e^(-bt/2m). Optimal for instruments needing quick settling.

### Slide 8: Systems of ODEs

- Multiple coupled equations arise naturally. Any nth-order ODE can be rewritten as a first-order system.
- dx/dt = Ax + b(t), where x is a vector and A is a matrix
- For constant A, the solution involves the matrix exponential: x(t) = e^(At) * x(0). Eigenvalues of A determine behavior:
- All eigenvalues with negative real part: stable node/spiral (decays to 0)
- All positive real part: unstable (solutions diverge)
- Pure imaginary: center (neutrally stable oscillation)
- Mixed signs: saddle point (unstable)
- Phase portraits visualize trajectories in state space

### Slide 9: Phase Plane Analysis

- For 2D autonomous systems x' = f(x,y), y' = g(x,y), the phase plane reveals qualitative behavior without solving explicitly.
- Equilibria
- Points where f = g = 0. Classified by eigenvalues of Jacobian: nodes, spirals, saddles, centers. Stability determined by sign of real parts.
- Nullclines
- Curves where f=0 (horizontal flow) or g=0 (vertical flow). Intersections are equilibria. Flow direction between nullclines reveals global dynamics.
- Limit Cycles
- Isolated periodic orbits. Poincare-Bendixson theorem: bounded planar trajectories must approach equilibrium or limit cycle. Van der Pol oscillator is classic example.

### Slide 10: Laplace Transform

- Converts differential equations into algebraic equations. Especially powerful for initial-value problems with discontinuous forcing.
- L{f(t)} = F(s) = integral_0^inf e^(-st) f(t) dt
- Key Properties
- L{f'} = sF(s) - f(0): derivatives become polynomials in s
- L{e^(at)f} = F(s-a): frequency shifting
- L{f*g} = F(s)*G(s): convolution becomes multiplication
- L{delta(t-a)} = e^(-as): impulse response
- Solution Method
- Transform the ODE (algebraic in s)
- Solve for Y(s) using algebra
- Partial fractions decomposition
- Inverse transform to get y(t)

### Slide 11: Series Solutions and Special Functions

- When coefficients are not constant, power series methods generate solutions term by term, often yielding named special functions.
- Bessel's equation: x^2*y'' + x*y' + (x^2 - n^2)*y = 0. Solutions: J_n(x), Y_n(x). Arise in cylindrical geometries (drum vibrations, electromagnetic waveguides).
- Legendre's equation: (1-x^2)*y'' - 2x*y' + l(l+1)*y = 0. Legendre polynomials P_l(x). Spherical harmonics in quantum mechanics.
- Hermite equation: y'' - 2x*y' + 2n*y = 0. Hermite polynomials H_n. Quantum harmonic oscillator wavefunctions.
- Airy equation: y'' - x*y = 0. Airy functions Ai, Bi. Quantum tunneling, optics near caustics.
- Hypergeometric equation: unifies many special functions into a single framework (Gauss, 1812).

### Slide 12: Sturm-Liouville Theory

- The spectral theory of second-order linear operators. Provides the mathematical foundation for separation of variables in PDEs.
- d/dx[p(x)*y'] + [q(x) + lambda*w(x)]*y = 0 with boundary conditions
- Eigenvalues lambda_n form an infinite increasing sequence approaching infinity
- Eigenfunctions y_n form a complete orthogonal set (generalized Fourier series)
- Any "nice" function can be expanded: f(x) = sum c_n * y_n(x)
- Fourier series is the special case p=1, q=0, w=1 on [0,L]
- Foundation of quantum mechanics: Schrodinger equation is a Sturm-Liouville problem

### Slide 13: Partial Differential Equations

- PDEs involve multiple independent variables. The "big three" of classical mathematical physics:
- Heat Equation (Parabolic)
- u_t = k * u_xx. Diffusion, smoothing. Solution: initial temperature distribution smooths out exponentially. Solved by Fourier (1807).
- Wave Equation (Hyperbolic)
- u_tt = c^2 * u_xx. Propagation at finite speed c. D'Alembert solution: u = f(x-ct) + g(x+ct). Vibrating strings, sound, electromagnetics.
- Laplace Equation (Elliptic)
- u_xx + u_yy = 0. Steady-state. Harmonic functions satisfy maximum principle. Electrostatics, fluid flow, gravitational potential.

### Slide 14: Separation of Variables

- The most powerful elementary method for linear PDEs. Assume u(x,t) = X(x)*T(t) and separate into ODEs.
- Substitute product form into PDE
- Separate: each side depends on one variable only = constant (lambda)
- Solve resulting ODEs (often Sturm-Liouville problems)
- Apply boundary conditions to determine eigenvalues
- Superpose: u = sum c_n * X_n(x) * T_n(t)
- Use initial conditions to determine coefficients c_n (Fourier coefficients)
- Works for heat, wave, and Laplace equations on regular geometries (rectangles, circles, spheres).

### Slide 15: Fourier Series and Transforms

- Joseph Fourier's 1807 insight: any periodic function can be decomposed into sines and cosines. This revolutionized both mathematics and physics.
- Fourier Series
- f(x) = a_0/2 + sum[a_n*cos(nx) + b_n*sin(nx)]
- Coefficients: projections onto basis functions
- Convergence: pointwise for piecewise smooth f
- Gibbs phenomenon at discontinuities (~9% overshoot)
- Fourier Transform
- F(omega) = integral f(t)*e^(-i*omega*t) dt
- Extends to non-periodic functions on R
- Convolution theorem: F(f*g) = F(f)*F(g)
- Parseval's: energy in time = energy in frequency
- Uncertainty principle: narrow in time = wide in frequency

### Slide 16: The Navier-Stokes Equations

- The governing equations of fluid mechanics. One of the Clay Millennium Prize Problems ($1M) -- existence and smoothness of solutions in 3D remains unproven.
- rho*(du/dt + u*grad(u)) = -grad(p) + mu*laplacian(u) + f
- Nonlinear PDE system coupling velocity u, pressure p, and density rho
- Conservation of momentum + incompressibility constraint (div u = 0)
- Reynolds number Re = rho*U*L/mu determines laminar vs. turbulent flow
- Turbulence: still no complete mathematical theory. Kolmogorov's 1941 scaling law is empirical.
- Computational fluid dynamics (CFD) solves approximately on grids. Weather prediction, aircraft design, blood flow.

### Slide 17: Nonlinear Dynamics and Chaos

- Small changes in initial conditions can lead to wildly different outcomes. Deterministic systems can behave unpredictably.
- Lorenz System (1963)
- x'=sigma(y-x), y'=x(rho-z)-y, z'=xy-beta*z. Discovered sensitive dependence while modeling atmospheric convection. The "butterfly effect."
- Strange Attractors
- Fractal structures in phase space that attract trajectories. Lorenz attractor has Hausdorff dimension ~2.06. Bounded but never repeating.
- Lyapunov Exponents
- Quantify rate of divergence of nearby trajectories. Positive exponent = chaos. Sum of all exponents negative = dissipative system (attractor exists).

### Slide 18: Bifurcation Theory

- How do solutions change qualitatively as a parameter varies? Bifurcation points mark transitions between different dynamical regimes.
- Saddle-node bifurcation: two equilibria collide and annihilate. x' = r + x^2. Equilibria exist only for r Transcritical bifurcation: equilibria exchange stability. x' = rx - x^2. Origin stable for r 0.
- Pitchfork bifurcation: symmetry-breaking. x' = rx - x^3. One equilibrium splits into three at r = 0.
- Hopf bifurcation: equilibrium loses stability, periodic orbit born. Models onset of oscillation in circuits, chemical reactions, ecology.
- Period-doubling cascade: route to chaos. Successive doublings of period as parameter changes. Universal Feigenbaum constants (delta = 4.669...).

### Slide 19: Numerical Methods for ODEs

- Most differential equations have no closed-form solution. Numerical methods approximate solutions on discrete time steps.
- MethodOrderProperties
- Euler (Forward)1Simplest. Error O(h). Often unstable for stiff problems.
- Midpoint (RK2)2One midpoint evaluation. Error O(h^2). Better accuracy.
- Classical RK44Four evaluations per step. Error O(h^4). Workhorse method.
- Dormand-Prince (RK45)4-5Adaptive step size. Embedded pair. MATLAB ode45 default.
- Adams-BashforthkMultistep. Reuses past evaluations. Efficient per step.
- BDF (Backward Diff)kImplicit. Stable for stiff systems. MATLAB ode15s.

### Slide 20: Stiff Systems

- A system is "stiff" when solution components evolve on vastly different timescales. Explicit methods require impractically small steps.
- The Problem
- Fast transients decay quickly but force tiny step sizes
- Explicit Euler on y'=-1000y with h>0.002 is unstable
- Chemical kinetics: reactions span 10^-12 to 10^3 seconds
- Circuit simulation: fast switching + slow RC decay
- The Solution
- Implicit methods: solve nonlinear systems each step
- BDF methods (up to order 6): A-stable or stiffly stable
- Implicit Runge-Kutta: RADAU, SDIRK methods
- Exponential integrators: exact for linear part
- Cost per step higher, but enormously fewer steps needed

### Slide 21: Numerical Methods for PDEs

- Finite Differences (FDM)
- Replace derivatives with discrete approximations on a grid. u_xx ~ (u_{i+1} - 2u_i + u_{i-1})/h^2. Simple, flexible, widely used for regular geometries.
- Finite Elements (FEM)
- Decompose domain into triangles/tetrahedra. Approximate solution as sum of basis functions. Handles complex geometry. Dominant in structural mechanics.
- Spectral Methods
- Expand solution in global basis (Fourier, Chebyshev). Exponential convergence for smooth solutions. Used in weather/climate models and turbulence DNS.
- Finite Volumes (FVM)
- Conserve fluxes across cell boundaries. Natural for conservation laws. Dominant in CFD (fluid dynamics). Handles shocks via Riemann solvers.

### Slide 22: Maxwell's Equations

- James Clerk Maxwell's 1865 PDEs unify electricity, magnetism, and optics. They predicted electromagnetic waves traveling at the speed of light.
- curl E = -dB/dt, curl B = mu_0*J + mu_0*epsilon_0*dE/dt, div E = rho/epsilon_0, div B = 0
- Four coupled PDEs relating electric field E and magnetic field B
- In vacuum: wave equation for E and B with speed c = 1/sqrt(mu_0*epsilon_0)
- Predicted radio waves (verified by Hertz, 1887)
- Special relativity emerges from their Lorentz invariance
- Numerical solution: FDTD (Yee's algorithm, 1966) used in antenna and chip design

### Slide 23: The Schrodinger Equation

- The fundamental equation of quantum mechanics (1926). A PDE governing the wave function psi(x,t) of a quantum system.
- i*hbar * dpsi/dt = -(hbar^2/2m) * d^2psi/dx^2 + V(x)*psi
- Linear PDE: superposition principle holds (quantum superposition)
- Time-independent form: eigenvalue problem for energy levels E_n
- Hydrogen atom: exact solution yields spectral lines (Bohr's model explained)
- Harmonic oscillator: equally spaced energy levels (phonons, photons)
- Tunneling: classically forbidden regions have exponentially decaying psi
- Many-body version: N particles in 3D requires solving in 3N dimensions

### Slide 24: Einstein's Field Equations

- General relativity (1915): gravity is the curvature of spacetime, described by a system of 10 coupled nonlinear PDEs.
- G_mu_nu + Lambda*g_mu_nu = (8*pi*G/c^4) * T_mu_nu
- G_mu_nu (Einstein tensor) encodes spacetime curvature
- T_mu_nu (stress-energy tensor) encodes matter/energy distribution
- Exact solutions: Schwarzschild (black holes), Friedmann (expanding universe), Kerr (rotating black holes)
- Numerical relativity: simulate binary black hole mergers (gravitational waves detected 2015)
- The cosmological constant Lambda: dark energy driving accelerated expansion

### Slide 25: Biological Applications

- Population Dynamics
- Lotka-Volterra: predator-prey cycles. Logistic growth: dN/dt = rN(1-N/K). SIR model: epidemic spread as coupled ODEs. Competitive exclusion principle.
- Reaction-Diffusion
- Turing patterns (1952): morphogenesis from instability in coupled diffusing chemicals. Explains animal coat patterns, coral structures, chemical oscillations.
- Neuroscience
- Hodgkin-Huxley model (1952, Nobel Prize): 4 coupled ODEs for action potential. FitzHugh-Nagumo simplification. Neural network dynamics.
- Cardiac Modeling
- Bidomain equations: coupled PDEs for electrical propagation in heart tissue. Used to understand arrhythmias and design defibrillators.

### Slide 26: Financial Mathematics

- Stochastic differential equations (SDEs) model financial markets, incorporating randomness through Brownian motion.
- Black-Scholes Equation (1973)
- dV/dt + (1/2)*sigma^2*S^2*d^2V/dS^2 + r*S*dV/dS - r*V = 0
- PDE for option pricing. Nobel Prize 1997. Assumes log-normal stock prices, constant volatility, no transaction costs.
- Stochastic Calculus
- Geometric Brownian motion: dS = mu*S*dt + sigma*S*dW
- Ito's lemma: chain rule for stochastic processes
- Risk-neutral pricing: change measure, discount at risk-free rate
- Monte Carlo simulation: sample paths of SDEs

### Slide 27: Variational Methods

- Many physical laws arise from minimizing a functional (action, energy). The Euler-Lagrange equation bridges variational principles and differential equations.
- Euler-Lagrange: d/dt(dL/dq') - dL/dq = 0 for Lagrangian L = T - V
- Hamilton's principle: true trajectory minimizes the action S = integral L dt
- Brachistochrone problem (1696): curve of fastest descent is a cycloid
- Geodesics: shortest paths on curved surfaces satisfy E-L equations
- Noether's theorem (1918): every continuous symmetry gives a conservation law (time->energy, space->momentum, rotation->angular momentum)
- Modern physics: Standard Model Lagrangian encodes all known particle interactions

### Slide 28: Historical Timeline

- 1687
- Newton's Principia: F = ma is a second-order ODE. Calculus invented (simultaneously by Leibniz) to solve it.
- 1807
- Fourier presents heat equation solution via trigonometric series. Rejected initially, later transforms mathematics.
- 1822
- Cauchy proves existence theorems. Rigor enters the field.
- 1890
- Poincare's qualitative theory: phase portraits, index theory, chaos foreshadowed.
- 1926
- Schrodinger equation. Quantum mechanics formulated as a PDE eigenvalue problem.
- 1963
- Lorenz discovers deterministic chaos in a 3-ODE weather model.
- 2000
- Navier-Stokes existence and smoothness named Millennium Prize Problem.

### Slide 29: Modern Frontiers

- Machine Learning + DEs
- Neural ODEs (2018): continuous-depth networks. Physics-informed neural networks (PINNs) solve PDEs without grids. Learned simulators for turbulence.
- Geometric Integration
- Structure-preserving numerical methods. Symplectic integrators for Hamiltonian systems. Conserve energy over astronomical timescales.
- Infinite-Dimensional Dynamics
- PDEs as dynamical systems in function spaces. Attractor theory, inertial manifolds. Understanding turbulence through infinite-dimensional chaos.
- Quantum Computing for DEs
- HHL algorithm promises exponential speedup for linear systems. Variational quantum eigensolvers for molecular Schrodinger equation. Still early stage.

### Slide 30: Software Ecosystem

- ToolStrengths
- MATLAB / Simulinkode45/ode15s, block diagrams, industry standard for control systems
- Python (SciPy)solve_ivp, integrate.odeint, open source, ML integration
- Julia (DifferentialEquations.jl)Fastest general-purpose solver suite, 300+ algorithms, stiff/non-stiff
- Mathematica / MapleSymbolic solutions, DSolve/dsolve, visualization
- COMSOL / ANSYSMultiphysics FEM, industrial PDE simulation
- FEniCS / deal.IIOpen-source FEM frameworks for research
- OpenFOAMCFD (Navier-Stokes), open source, industrial adoption

### Slide 31: Key Takeaways

- Language of Nature
- Differential equations express the fundamental laws of physics, biology, and economics. They translate "rates of change" into precise predictions.
- Theory Meets Computation
- Existence theorems guarantee solutions exist; numerical methods let us compute them. The interplay drives modern applied mathematics.
- Chaos Is Deterministic
- Simple differential equations can produce unpredictable behavior. Chaos theory revealed that complexity needs not arise from complicated rules.
- Open Frontiers
- Millennium Prize Problems (Navier-Stokes), turbulence, neural DEs, and quantum simulation ensure the field remains vibrant and central to science.
- -- End --


## Related Decks

- [Topology](https://shipslides.com/d/mathematics-topology)
- [Algebra](https://shipslides.com/d/mathematics-algebra)
- [Applied Mathematics](https://shipslides.com/d/mathematics-applied-math)
- [Calculus](https://shipslides.com/d/mathematics-calculus)