➗Calculus II Unit 4 – Introduction to Differential Equations

What Are Differential Equations?

• Equations involving derivatives of one or more dependent variables with respect to one or more independent variables
• Describe the rate of change of a quantity in relation to another quantity
• Denoted by the order of the highest derivative present in the equation ($\frac{dy}{dx}$ for first-order, $\frac{d^2y}{dx^2}$ for second-order)
• Classified as ordinary differential equations (ODEs) when there is only one independent variable and partial differential equations (PDEs) when there are multiple independent variables
• Used to model various phenomena in physics, engineering, economics, and other fields (population growth, heat transfer, electrical circuits)
• Solutions to differential equations are functions that satisfy the given equation and any initial or boundary conditions
• Can be solved analytically using various techniques (separation of variables, integrating factors, power series) or numerically using computational methods (Euler's method, Runge-Kutta methods)

Types of Differential Equations

• Classified based on order, linearity, and homogeneity
• First-order differential equations contain only first derivatives ($\frac{dy}{dx} = f(x, y)$)
• Higher-order differential equations involve derivatives of order two or more ($\frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0$)
• Linear differential equations have the dependent variable and its derivatives appearing linearly, with no products or powers ($\frac{dy}{dx} + P(x)y = Q(x)$)
  • Homogeneous linear differential equations have $Q(x) = 0$
  • Non-homogeneous linear differential equations have $Q(x) \neq 0$
• Nonlinear differential equations have the dependent variable or its derivatives appearing as powers, products, or in other nonlinear forms ($\frac{dy}{dx} = y^2 + \sin(x)$)
• Exact differential equations can be written in the form $M(x, y)dx + N(x, y)dy = 0$, where $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$
• Separable differential equations can be written in the form $\frac{dy}{dx} = f(x)g(y)$, allowing variables to be separated and integrated

Solving First-Order Differential Equations

• Separation of variables involves rewriting the equation to separate the variables, then integrating both sides ($\frac{dy}{dx} = f(x)g(y) \rightarrow \int \frac{1}{g(y)}dy = \int f(x)dx$)
• Integrating factor method multiplies both sides of a linear equation by a function $\mu(x)$ to make the equation exact ($\mu(x)[\frac{dy}{dx} + P(x)y] = \mu(x)Q(x)$)
• Exact equations can be solved by finding a potential function $\Psi(x, y)$ such that $\frac{\partial \Psi}{\partial x} = M(x, y)$ and $\frac{\partial \Psi}{\partial y} = N(x, y)$
• Bernoulli equations of the form $\frac{dy}{dx} + P(x)y = Q(x)y^n$ can be transformed into linear equations by substituting $v = y^{1-n}$
• Homogeneous equations of the form $\frac{dy}{dx} = f(\frac{y}{x})$ can be solved by substituting $v = \frac{y}{x}$
• Numerical methods, such as Euler's method or Runge-Kutta methods, approximate solutions by iteratively calculating the function value at discrete points

Applications of First-Order DEs

• Exponential growth and decay models (population growth, radioactive decay)
  • $\frac{dP}{dt} = kP$, where $P$ is the population size and $k$ is the growth rate
  • $\frac{dN}{dt} = -\lambda N$, where $N$ is the amount of a radioactive substance and $\lambda$ is the decay constant
• Mixing problems involving the concentration of a substance in a tank with inflow and outflow (salt water mixing)
  • $\frac{dA}{dt} = r_iC_i - r_oC$, where $A$ is the amount of substance, $r_i$ and $r_o$ are the inflow and outflow rates, and $C_i$ and $C$ are the inflow and tank concentrations
• Newton's law of cooling describes the temperature change of an object in a surrounding medium ($\frac{dT}{dt} = -k(T - T_m)$, where $T$ is the object's temperature, $T_m$ is the medium's temperature, and $k$ is a constant)
• Torricelli's law for fluid flow through an orifice ($\frac{dh}{dt} = -k\sqrt{h}$, where $h$ is the height of the fluid and $k$ is a constant)
• Logistic population growth models limited growth with a carrying capacity ($\frac{dP}{dt} = kP(1 - \frac{P}{K})$, where $K$ is the carrying capacity)
• Kirchhoff's laws for electrical circuits (sum of currents entering a node equals sum of currents leaving, sum of voltage drops around a loop equals zero)

Higher-Order Differential Equations

• Require additional initial or boundary conditions to determine a unique solution
• Linear higher-order equations with constant coefficients can be solved using the characteristic equation ($ar^n + br^{n-1} + \cdots + k = 0$)
  • Distinct real roots lead to solutions of the form $y = c_1e^{r_1x} + c_2e^{r_2x} + \cdots + c_ne^{r_nx}$
  • Complex conjugate roots lead to solutions with trigonometric functions ($y = e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$)
  • Repeated roots require solutions with polynomial terms ($y = (c_1 + c_2x + \cdots + c_mx^{m-1})e^{rx}$)
• Reduction of order method reduces the order of the equation by substituting $y = uv$, where $u$ is a known solution and $v$ is a new function to be determined
• Variation of parameters method finds a particular solution to a non-homogeneous equation by replacing constants with functions in the general solution of the corresponding homogeneous equation
• Series solutions express the solution as an infinite power series ($y = \sum_{n=0}^{\infty} a_nx^n$) and determine the coefficients by substituting the series into the differential equation
• Cauchy-Euler equations have variable coefficients of the form $x^n$ ($x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0$) and can be solved using the substitution $x = e^t$

Laplace Transforms

• Integral transform that converts a differential equation into an algebraic equation
• Laplace transform of a function $f(t)$ is defined as $\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st}f(t)dt$
• Linearity property: $\mathcal{L}\{af(t) + bg(t)\} = a\mathcal{L}\{f(t)\} + b\mathcal{L}\{g(t)\}$
• Differentiation property: $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$, $\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0)$
• Integration property: $\mathcal{L}\{\int_0^t f(\tau)d\tau\} = \frac{1}{s}F(s)$
• Shifting property: $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$
• Convolution property: $\mathcal{L}\{(f * g)(t)\} = F(s)G(s)$, where $(f * g)(t) = \int_0^t f(\tau)g(t-\tau)d\tau$
• Solving differential equations using Laplace transforms:
  1. Take the Laplace transform of the differential equation and initial conditions
  2. Solve the resulting algebraic equation for $F(s)$
  3. Find the inverse Laplace transform of $F(s)$ to obtain the solution $f(t)$

Systems of Differential Equations

• Involve multiple dependent variables and their derivatives
• Can be written in vector form $\frac{d\vec{x}}{dt} = \vec{f}(\vec{x}, t)$, where $\vec{x}$ is a vector of dependent variables and $\vec{f}$ is a vector-valued function
• Linear systems have the form $\frac{d\vec{x}}{dt} = A\vec{x} + \vec{b}(t)$, where $A$ is a matrix of constants and $\vec{b}(t)$ is a vector-valued function
  • Homogeneous linear systems have $\vec{b}(t) = \vec{0}$
  • Non-homogeneous linear systems have $\vec{b}(t) \neq \vec{0}$
• Eigenvalues and eigenvectors of the matrix $A$ determine the behavior of the system
  • Real, distinct eigenvalues lead to exponential solutions
  • Complex conjugate eigenvalues lead to oscillatory solutions
  • Repeated eigenvalues may require generalized eigenvectors and polynomial terms in the solution
• The Jacobian matrix $J(\vec{x})$ of a nonlinear system $\frac{d\vec{x}}{dt} = \vec{f}(\vec{x})$ determines the local stability of equilibrium points
  • If all eigenvalues of $J(\vec{x}^*)$ have negative real parts, the equilibrium point $\vec{x}^*$ is asymptotically stable
  • If any eigenvalue has a positive real part, the equilibrium point is unstable
  • If all eigenvalues have zero real parts, further analysis is required to determine stability

Real-World Applications

• Population dynamics models interactions between multiple species (predator-prey, competition, symbiosis)
  • Lotka-Volterra equations: $\frac{dx}{dt} = ax - bxy$, $\frac{dy}{dt} = cxy - dy$, where $x$ and $y$ are prey and predator populations
• Epidemiology models the spread of infectious diseases (SIR model: Susceptible, Infected, Recovered)
  • $\frac{dS}{dt} = -\beta SI$, $\frac{dI}{dt} = \beta SI - \gamma I$, $\frac{dR}{dt} = \gamma I$, where $\beta$ is the infection rate and $\gamma$ is the recovery rate
• Pharmacokinetics describes the absorption, distribution, metabolism, and excretion of drugs in the body
  • One-compartment model: $\frac{dC}{dt} = -kC$, where $C$ is the drug concentration and $k$ is the elimination rate constant
• Chemical kinetics models the rates of chemical reactions (first-order, second-order, enzyme kinetics)
  • First-order reaction: $\frac{d[A]}{dt} = -k[A]$, where $[A]$ is the concentration of reactant $A$ and $k$ is the rate constant
• Heat and mass transfer problems involve the diffusion of heat or substances (Fourier's law, Fick's law)
  • One-dimensional heat equation: $\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}$, where $T$ is temperature and $\alpha$ is the thermal diffusivity
• Fluid dynamics models the flow of liquids and gases (Navier-Stokes equations)
  • Continuity equation: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0$, where $\rho$ is the fluid density and $\vec{v}$ is the velocity field
• Quantum mechanics describes the behavior of particles at the atomic and subatomic scale (Schrödinger equation)
  • Time-dependent Schrödinger equation: $i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi + V\Psi$, where $\Psi$ is the wave function, $\hbar$ is the reduced Planck's constant, $m$ is the particle mass, and $V$ is the potential energy