♾️AP Calculus AB/BC Unit 7 – Differential Equations

What are Differential Equations?

• Equations that involve derivatives of an unknown function
• Relate a function to its derivatives
• Describe how a system changes over time
• Used to model real-world phenomena in various fields (physics, engineering, economics, biology)
• Classified based on order, linearity, and number of variables
  • Order determined by the highest derivative present
  • Linear equations have the unknown function and its derivatives appear linearly
  • Ordinary differential equations (ODEs) involve a single independent variable
  • Partial differential equations (PDEs) involve multiple independent variables

Types of Differential Equations

• First-order differential equations
  • Involve only the first derivative of the unknown function
  • Examples: $\frac{dy}{dx} = f(x, y)$, $y' + P(x)y = Q(x)$
• Higher-order differential equations
  • Involve derivatives of order two or higher
  • Example: $\frac{d^2y}{dx^2} + a\frac{dy}{dx} + by = f(x)$
• Linear differential equations
  • Unknown function and its derivatives appear linearly
  • Can be homogeneous or non-homogeneous
    • Homogeneous: right-hand side is zero
    • Non-homogeneous: right-hand side is a non-zero function
• Nonlinear differential equations
  • Unknown function or its derivatives appear in a nonlinear manner
  • Example: $\frac{dy}{dx} = y^2 + \sin(x)$
• Ordinary differential equations (ODEs)
  • Involve a single independent variable
  • Example: $\frac{d^2x}{dt^2} + kx = 0$
• Partial differential equations (PDEs)
  • Involve multiple independent variables
  • Example: $\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0$ (Laplace's equation)

Solving First-Order Differential Equations

• Separation of variables
  • Applicable when the equation can be written as $\frac{dy}{dx} = f(x)g(y)$
  • Separate variables and integrate both sides
• Integrating factor method
  • Used for linear first-order equations of the form $\frac{dy}{dx} + P(x)y = Q(x)$
  • Multiply both sides by an integrating factor to make the left-hand side a total derivative
• Exact equations
  • Equation of the form $M(x, y)dx + N(x, y)dy = 0$
  • Condition for exactness: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$
  • Solve by finding a potential function $\phi(x, y)$ such that $\frac{\partial \phi}{\partial x} = M$ and $\frac{\partial \phi}{\partial y} = N$
• Bernoulli equations
  • Nonlinear equations of the form $\frac{dy}{dx} + P(x)y = Q(x)y^n$
  • Substitute $z = y^{1-n}$ to transform into a linear equation
• Homogeneous equations
  • Equation of the form $\frac{dy}{dx} = f(\frac{y}{x})$
  • Substitute $u = \frac{y}{x}$ to reduce the order of the equation

Applications of Differential Equations

• Population growth models
  • Exponential growth: $\frac{dP}{dt} = kP$
  • Logistic growth: $\frac{dP}{dt} = kP(1 - \frac{P}{K})$
• Radioactive decay
  • First-order decay: $\frac{dN}{dt} = -\lambda N$
  • Half-life: $t_{1/2} = \frac{\ln(2)}{\lambda}$
• Cooling and heating problems
  • Newton's law of cooling: $\frac{dT}{dt} = -k(T - T_a)$
• Mechanical vibrations
  • Mass-spring system: $m\frac{d^2x}{dt^2} + kx = 0$
  • Damped oscillations: $m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0$
• Electrical circuits
  • RC circuit: $RC\frac{dV}{dt} + V = V_0$
  • RLC circuit: $L\frac{d^2I}{dt^2} + R\frac{dI}{dt} + \frac{1}{C}I = 0$
• Fluid dynamics
  • Torricelli's law: $\frac{dV}{dt} = -A\sqrt{2gh}$

Higher-Order Differential Equations

• Linear equations with constant coefficients
  • Characteristic equation: $ar^2 + br + c = 0$ for $ay'' + by' + cy = 0$
  • Solutions based on roots of the characteristic equation
    • Distinct real roots: $y = c_1e^{r_1x} + c_2e^{r_2x}$
    • Repeated real roots: $y = (c_1 + c_2x)e^{rx}$
    • Complex conjugate roots: $y = e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$
• Non-homogeneous equations
  • Particular solution: found using undetermined coefficients or variation of parameters
  • General solution: sum of the complementary solution (homogeneous) and particular solution
• Cauchy-Euler equations
  • Equations of the form $ax^2y'' + bxy' + cy = 0$
  • Substitute $x = e^t$ to transform into a linear equation with constant coefficients
• Series solutions
  • Assume a power series solution: $y = \sum_{n=0}^{\infty} a_nx^n$
  • Determine the coefficients by substituting the series into the differential equation
• Laplace transforms
  • Transform a differential equation into an algebraic equation
  • Solve the algebraic equation and apply the inverse Laplace transform to obtain the solution

Systems of Differential Equations

• Coupled equations involving multiple unknown functions
• Example: predator-prey model (Lotka-Volterra equations)
  • $\frac{dx}{dt} = ax - bxy$
  • $\frac{dy}{dt} = cxy - dy$
• Solve by eliminating one variable or using matrix methods
  • Eigenvalues and eigenvectors for linear systems with constant coefficients
• Phase plane analysis
  • Visualize the behavior of solutions in the xy-plane
  • Identify equilibrium points and their stability
• Linearization
  • Approximate a nonlinear system near an equilibrium point using a linear system
  • Determine the stability of the equilibrium point based on the eigenvalues of the linearized system

Numerical Methods for Differential Equations

• Euler's method
  • First-order approximation: $y_{n+1} = y_n + hf(x_n, y_n)$
  • Improved Euler's method: $y_{n+1} = y_n + \frac{h}{2}(f(x_n, y_n) + f(x_{n+1}, y_n + hf(x_n, y_n)))$
• Runge-Kutta methods
  • Higher-order approximations
  • Fourth-order Runge-Kutta (RK4): $y_{n+1} = y_n + \frac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4)$
• Multistep methods
  • Use information from previous steps to approximate the solution
  • Adams-Bashforth methods: explicit
  • Adams-Moulton methods: implicit
• Stability and convergence
  • Stability: numerical solution remains bounded as the step size decreases
  • Convergence: numerical solution approaches the exact solution as the step size decreases
• Adaptive step size control
  • Adjust the step size based on the estimated error to maintain accuracy and efficiency

Key Theorems and Concepts

• Existence and uniqueness theorem
  • Guarantees the existence and uniqueness of a solution to an initial value problem (IVP)
  • Requires the right-hand side of the differential equation to be continuous and Lipschitz continuous
• Picard's iteration
  • Constructive method to prove the existence and uniqueness of a solution to an IVP
  • Generates a sequence of functions that converges to the solution
• Gronwall's inequality
  • Estimates the growth of a function satisfying a certain integral inequality
  • Used to prove the continuous dependence of solutions on initial conditions and parameters
• Sturm-Liouville theory
  • Deals with eigenvalue problems for second-order linear differential equations
  • Orthogonality of eigenfunctions
  • Completeness of the set of eigenfunctions
• Green's functions
  • Used to solve non-homogeneous linear differential equations with specified boundary conditions
  • Represents the impulse response of the system
• Lyapunov stability theory
  • Analyzes the stability of equilibrium points in nonlinear systems
  • Lyapunov functions: scalar functions that decrease along the trajectories of the system
• Poincaré-Bendixson theorem
  • Classifies the possible behaviors of solutions to planar autonomous systems
  • Limit cycles, periodic orbits, and chaos