Comprehensive Study Notes: Introduction to Differential Equations

Introduction to Differential Equations

Basic Definitions and Concepts

  • Differential Equation (DE): Any equation containing the derivatives of one or more dependent variables with respect to one or more independent variables.
  • Fundamental Problem: Given a differential equation, how does one solve for the unknown function $y = f(x)$? This is essentially the reverse problem of differential calculus (finding an antiderivative).
  • Solution of an ODE: Any function $f$, defined on an interval $I$ and possessing at least $n$ derivatives continuous on $I$, which reduces the $n$th-order differential equation to an identity upon substitution.
  • Trivial Solution: A solution that is identically zero on an interval $I$ ($y = 0$).

Classification of Differential Equations

By Type
  • Ordinary Differential Equation (ODE): Contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable.
    • Example: d2ydx2+dydx+12y=0\frac{d^2y}{dx^2} + \frac{dy}{dx} + 12y = 0
  • Partial Differential Equation (PDE): Involves partial derivatives of one or more dependent variables of two or more independent variables.
    • Example: ∂2u∂x2+∂2u∂y2=0\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
By Order
  • The order of a differential equation is the order of the highest derivative in the equation.
  • First-Order ODE General Form: $F(x, y, y') = 0$.
  • $n$th-Order ODE General Form: $F(x, y, y', \dots, y^{(n)}) = 0$.
  • Normal Form: An equation solved uniquely for the highest derivative: dnydxn=f(x,y,y′,…,y(n−1))\frac{d^ny}{dx^n} = f(x, y, y', \dots, y^{(n-1)}).
By Linearity

An $n$th-order ODE is linear if $F$ is linear in $y, y', \dots, y^{(n)}$. It must satisfy two properties:

  1. The dependent variable $y$ and its derivatives are all of the first degree (power is 1).
  2. The coefficients $a0, a1, \dots, a_n$ depend at most on the independent variable $x$.
  • General Linear Form: a<em>n(x)y(n)+a</em>n−1(x)y(n−1)+⋯+a<em>1(x)y′+a</em>0(x)y=g(x)a<em>n(x) y^{(n)} + a</em>{n-1}(x) y^{(n-1)} + \dots + a<em>1(x) y' + a</em>0(x) y = g(x).
  • Nonlinear Equation: An equation that is not linear (e.g., contains $y^2$, $\sin y$, or $e^{y'}$).

Solutions and Intervals

  • Interval of Definition: Also called the interval of validity or domain of the solution. It is the interval $I$ (open, closed, or infinite) on which the solution function is defined and differentiable.
  • Solution Curve: The graph of a solution $f$ of an ODE. There is a distinction between the domain of the function $f$ and the interval of definition of the solution $f$.
  • Explicit Solution: The dependent variable is expressed solely in terms of the independent variable and constants (e.g., $y = f(x)$).
  • Implicit Solution: A relation $G(x, y) = 0$ that defines one or more explicit solutions on an interval $I$.
  • Particular Solution: A solution free of arbitrary parameters.
  • Singular Solution: A solution that cannot be obtained by specializing the parameters in a family of solutions.

Initial-Value Problems (IVP)

  • Definition: A problem consisting of an $n$th-order differential equation subject to $n$ side conditions specified at a single point $x_0$ (initial conditions).
  • First-Order IVP: Solve $y' = f(x, y)$ subject to $y(x0) = y0$.
  • Second-Order IVP: Solve $y'' = f(x, y, y')$ subject to $y(x0) = y0, y'(x0) = y1$.
  • Geometrically: A first-order IVP seeks a curve passing through a point. A second-order IVP seeks a curve passing through a point with a specific slope.
  • Theorem 1.2.1 (Existence and Uniqueness): Let $R$ be a rectangular region containing $(x0, y0)$. If $f(x, y)$ and ∂f∂y\frac{\partial f}{\partial y} are continuous on $R$, then there exists a unique solution $y(x)$ on some interval $I_0$ contained in $R$.

Differential Equations as Mathematical Models

Mathematical modeling involves describing physical behavior in mathematical terms.

Population Dynamics
  • Malthusian Model: dPdt=kP\frac{dP}{dt} = kP, where $P$ is population and $k$ is a constant of proportionality. It assumes the rate of growth is proportional to the current population.
Radioactive Decay
  • Model: dAdt=kA\frac{dA}{dt} = kA. Similar to population growth, but $k < 0$ for decay.
Newton's Law of Cooling/Warming
  • Model: dTdt=k(T−T<em>m)\frac{dT}{dt} = k(T - T<em>m), where $T$ is the temperature of the object and $Tm$ is the ambient temperature of the surrounding medium.
Spread of Disease
  • Model: dxdt=kx(n+1−x)\frac{dx}{dt} = kx(n + 1 - x), where $x$ is the number of infected people in a community of $n$ people.
Chemical Reactions
  • First-Order: dXdt=kX\frac{dX}{dt} = kX.
  • Second-Order: dXdt=k(a−X)(b−X)\frac{dX}{dt} = k(a - X)(b - X).
Series Circuits
  • LRC-Series Circuit: Governed by $L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C} q = E(t)$, where $q$ is charge, $L$ is inductance, $R$ is resistance, and $C$ is capacitance.
Falling Bodies
  • No Air Resistance: d2sdt2=−g\frac{d^2s}{dt^2} = -g.
  • With Air Resistance: $m \frac{dv}{dt} = mg - kv$, where $k$ is the drag coefficient.

Qualitative Analysis: Solution Curves Without a Solution

Direction Fields
  • Lineal Element: A small line segment representing the slope $f(x, y)$ at a point $(x, y)$.
  • Direction Field: A collection of lineal elements that suggests the overall shape of solution curves.
Autonomous First-Order DEs
  • Definition: An ODE where the independent variable $x$ does not appear explicitly: dydx=f(y)\frac{dy}{dx} = f(y).
  • Critical Points: Zeros of $f(y)$ (where $f(c) = 0$). These correspond to constant/equilibrium solutions.
  • Phase Portrait: A vertical line (phase line) showing the critical points and arrows indicating if $y(x)$ is increasing or decreasing.
  • Stability:
    • Asymptotically Stable (Attractor): Arrows on both sides point toward the critical point.
    • Unstable (Repeller): Arrows on both sides point away from the critical point.
    • Semi-stable: Arrows on one side point toward and the other side point away.

Analytical Methods for First-Order DEs

Separable Equations
  • Form: dydx=g(x)h(y)\frac{dy}{dx} = g(x)h(y).
  • Method: Rewrite as 1h(y)dy=g(x)dx\frac{1}{h(y)} dy = g(x) dx and integrate both sides.
Linear Equations
  • Standard Form: dydx+P(x)y=f(x)\frac{dy}{dx} + P(x)y = f(x).
  • Integrating Factor: μ(x)=e∫P(x)dx\mu(x) = e^{\int P(x) dx}.
  • Method: Multiply the standard form by $\mu(x)$, recognize the left side as ddx[μ(x)y]\frac{d}{dx}[\mu(x)y], and integrate.
Exact Equations
  • General Form: $M(x, y) dx + N(x, y) dy = 0$.
  • Criterion for Exactness: ∂M∂y=∂N∂x\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.
  • Method: Find a potential function $f(x, y)$ such that ∂f∂x=M\frac{\partial f}{\partial x} = M and ∂f∂y=N\frac{\partial f}{\partial y} = N. The solution is $f(x, y) = c$.
Bernoulli's Equation
  • Form: dydx+P(x)y=f(x)yn\frac{dy}{dx} + P(x)y = f(x)y^n ($n \neq 0, 1$).
  • Substitution: $u = y^{1-n}$ reduces it to a linear equation.

Numerical Methods

Euler's Method
  • Algorithm: $y{n+1} = yn + h f(xn, yn)$, where $h$ is the step size.
  • Concept: Uses successive tangent line segments to approximate a solution curve.

Higher-Order Linear Differential Equations

Homogeneous Equations
  • Superposition Principle: If $y1, y2, \dots, yk$ are solutions, then $y = c1y1 + c2y2 + \dots + cky_k$ is also a solution.
  • Linear Independence: A set of functions is independent if the only constants satisfying $c1f1 + \dots + cnfn = 0$ are $c_i = 0$ for all $i$.
  • Wronskian: A determinant $W(y1, \dots, yn)$ used to check for linear independence of solutions.
Constant Coefficients (Auxiliary Equation)
  • For $ay'' + by' + cy = 0$, solve $am^2 + bm + c = 0$:
    1. Distinct Real Roots: $y = c1e^{m1x} + c2e^{m2x}$.
    2. Repeated Real Roots: $y = c1e^{m1x} + c2xe^{m1x}$.
    3. Complex Roots ($a \pm i\beta$): $y = e^{ax}(c1 \cos \beta x + c2 \sin \beta x)$.
Undetermined Coefficients
  • Used for nonhomogeneous equations $ay^{(n)} + \dots + a_0y = g(x)$ where $g(x)$ contains polynomials, exponentials, or sines/cosines.
  • Requires assuming a trial solution $y_p$ based on $g(x)$.
Variation of Parameters
  • A more general method for finding $y_p$. For second-order equations:
    • $yp = u1y1 + u2y2$, where $u1' = \frac{W1}{W}$ and $u2' = \frac{W_2}{W}$.
Cauchy-Euler Equations
  • Form: $ax^2 y'' + bxy' + cy = g(x)$.
  • Substitution: $y = x^m$ leads to the auxiliary equation $am(m-1) + bm + c = 0$.

Laplace Transforms

  • Definition: Lf(t)=∫0∞e−stf(t)dt\mathcal{L}{f(t)} = \int_0^{\infty} e^{-st} f(t) dt.
  • Linearly: The transform of a linear combination is the linear combination of the transforms.
  • Inverse Transform: L−1F(s)=f(t)\mathcal{L}^{-1}{F(s)} = f(t).
  • Derivative Property: Lf′(t)=sF(s)−f(0)\mathcal{L}{f'(t)} = sF(s) - f(0).
  • Convolution Theorem: Lf∗g=F(s)G(s)\mathcal{L}{f * g} = F(s)G(s), where $f * g = \int_0^t f(\tau)g(t - \tau) d\tau$.