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: $\frac{d^2y}{dx^2} + \frac{dy}{dx} + 12y = 0$
• Partial Differential Equation (PDE): Involves partial derivatives of one or more dependent variables of two or more independent variables.
  • Example: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

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: $\frac{d^ny}{dx^n} = f(x, y, y', \dots, y^{(n-1)})$.

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 $a<em>0, a</em>1, \dots, a_n$ depend at most on the independent variable $x$.

• General Linear Form: $a<em>n(x) y^{(n)} + a</em>{n-1}(x) y^{(n-1)} + \dots + a<em>1(x) y' + a</em>0(x) y = g(x)$.
• 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 $y(x<em>0) = y</em>0$.
• Second-Order IVP: Solve $y'' = f(x, y, y')$ subject to $y(x<em>0) = y</em>0, y'(x<em>0) = y</em>1$.
• 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 $(x<em>0, y</em>0)$. If $f(x, y)$ and $\frac{\partial f}{\partial y}$ 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: $\frac{dP}{dt} = kP$, 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: $\frac{dA}{dt} = kA$. Similar to population growth, but $k < 0$ for decay.

Newton's Law of Cooling/Warming

• Model: $\frac{dT}{dt} = k(T - T<em>m)$, where $T$ is the temperature of the object and $T</em>m$ is the ambient temperature of the surrounding medium.

Spread of Disease

• Model: $\frac{dx}{dt} = kx(n + 1 - x)$, where $x$ is the number of infected people in a community of $n$ people.

Chemical Reactions

• First-Order: $\frac{dX}{dt} = kX$.
• Second-Order: $\frac{dX}{dt} = k(a - X)(b - X)$.

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: $\frac{d^2s}{dt^2} = -g$.
• 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: $\frac{dy}{dx} = f(y)$.
• 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: $\frac{dy}{dx} = g(x)h(y)$.
• Method: Rewrite as $\frac{1}{h(y)} dy = g(x) dx$ and integrate both sides.

Linear Equations

• Standard Form: $\frac{dy}{dx} + P(x)y = f(x)$.
• Integrating Factor: $\mu(x) = e^{\int P(x) dx}$.
• Method: Multiply the standard form by $\mu(x)$, recognize the left side as $\frac{d}{dx}[\mu(x)y]$, and integrate.

Exact Equations

• General Form: $M(x, y) dx + N(x, y) dy = 0$.
• Criterion for Exactness: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.
• Method: Find a potential function $f(x, y)$ such that $\frac{\partial f}{\partial x} = M$ and $\frac{\partial f}{\partial y} = N$. The solution is $f(x, y) = c$.

Bernoulli's Equation

• Form: $\frac{dy}{dx} + P(x)y = f(x)y^n$ ($n \neq 0, 1$).
• Substitution: $u = y^{1-n}$ reduces it to a linear equation.

Numerical Methods

Euler's Method

• Algorithm: $y<em>{n+1} = y</em>n + h f(x<em>n, y</em>n)$, 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 $y<em>1, y</em>2, \dots, y<em>k$ are solutions, then $y = c</em>1y<em>1 + c</em>2y<em>2 + \dots + c</em>ky_k$ is also a solution.
• Linear Independence: A set of functions is independent if the only constants satisfying $c<em>1f</em>1 + \dots + c<em>nf</em>n = 0$ are $c_i = 0$ for all $i$.
• Wronskian: A determinant $W(y<em>1, \dots, y</em>n)$ 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: $y = c<em>1e^{m</em>1x} + c<em>2e^{m</em>2x}$.
  2. Repeated Real Roots: $y = c<em>1e^{m</em>1x} + c<em>2xe^{m</em>1x}$.
  3. Complex Roots ($a \pm i\beta$): $y = e^{ax}(c<em>1 \cos \beta x + c</em>2 \sin \beta x)$.

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:
  • $y<em>p = u</em>1y<em>1 + u</em>2y<em>2$, where $u</em>1' = \frac{W<em>1}{W}$ and $u</em>2' = \frac{W_2}{W}$.

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