Differential Equations and Sequences & Series: Study Notes

Section 9.1 - Introduction to Differential Equations

Definitions and Concepts

Differential equations are equations involving derivatives of a function. They are fundamental in modeling various physical, biological, and economic systems.

• Differential Equation (DE): An equation involving derivatives of a function.

• Order: The highest derivative present in the equation.

• Initial Value Problem (IVP): A differential equation together with specified values at a given point.

• Linear Differential Equation: A DE in which the dependent variable and its derivatives appear to the first power and are not multiplied together.

• Equilibrium Solution: A constant solution where the derivative is zero.

• Stable/Unstable Equilibrium: Stability refers to whether solutions that start near an equilibrium stay near (stable) or move away (unstable).

General Solution

• Solving a DE generally means finding a function (or family of functions) that satisfies the equation.

• For an nth order DE, the general solution contains n arbitrary constants.

Initial Value Problems

• Given a DE and initial conditions, find the particular solution that satisfies both.

Verifying Solutions

• Substitute the proposed solution into the original DE to check if it holds.

Section 9.2 - Direction Fields and Euler's Method

Direction Fields

Direction fields (or slope fields) are graphical representations of first-order DEs, showing the slope of the solution curve at each point.

• Equilibrium Points: Points where the derivative is zero; solutions remain constant.

• Stability:

  • Stable: Solutions approach the equilibrium as time increases.

  • Unstable: Solutions move away from the equilibrium.

  • Semi-stable: Solutions approach from one side and move away from the other.

Euler's Method

Euler's Method is a numerical technique to approximate solutions to initial value problems.

• Given $\frac{dy}{dt} = f(t, y)$, $y(t_0) = y_0$, and step size $h$:

• Iterative formula: $y_{n+1} = y_n + h f(t_n, y_n)$

• Repeat for desired number of steps to approximate the solution.

Section 9.3 - Separable Differential Equations

Solving Separable Equations

A separable DE can be written as $\frac{dy}{dt} = g(t)h(y)$, allowing variables to be separated and integrated.

• Rewrite as $\frac{1}{h(y)} dy = g(t) dt$

• Integrate both sides to find the general solution.

Examples

• Verification: Given $u(t) = C_1 t^5 + C_2 t^{-4} - t^3$, verify it solves $t^2 u''(t) - 20u(t) = 14t^3$ by substitution and differentiation.

• Equilibrium Solutions: For $y'(t) = y(y-3)$, equilibrium at $y=0$ and $y=3$. For $y(0)=1$, the solution starts between equilibria and can be analyzed for stability.

• General Solution: For $y'(t) e^{t^2} = y^2 + 4$, separate variables and integrate.

• Other Examples:

  • $\frac{dy}{dx} = \frac{3y}{2x+1}$

  • $\frac{dy}{dx} = \frac{x^2 y - 4y}{x+2}$

  • $\frac{dy}{dx} = x e^{x^2} - \ln(y^2)$

Section 9.4 - Special First-Order Linear Differential Equations

Applications

• Harvesting: Models population with constant removal (harvesting) rate.

• Newton's Law of Cooling: $\frac{dT}{dt} = -k(T - T_{env})$, where $T$ is temperature, $T_{env}$ is ambient temperature, and $k$ is a positive constant.

Examples

• Loan Repayment: A student borrows $50,000 at 0.5% per month, pays $500/month. Model with$\frac{dP}{dt} = rP - 500$ and solve for time to repay.

• Weed Harvesting: Garlic Mustard Weed grows at 3% per month, 100 plants removed monthly, starting with 5,000. Model with $\frac{dN}{dt} = 0.03N - 100$ and solve for $N(t)$.

Section 9.5 - Applications of Differential Equations

Common Models

• Logistic Growth: $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$, where $K$ is carrying capacity.

• Gompertz Equation: $\frac{dP}{dt} = rP \ln\left(\frac{K}{P}\right)$, another model for bounded growth.

• Stirred Tank Reactions: Models concentration changes in a well-mixed tank with inflow and outflow.

Section 10.1 - Overview of Sequences and Series

Definitions

• Sequence: An ordered list of numbers, often defined by a formula $a_n$.

• Recursive Definition: Each term is defined in terms of previous terms.

• Explicit Definition: Each term is given directly as a function of $n$.

• Infinite Series: The sum of the terms of a sequence: $\sum_{n=1}^\infty a_n$.

• Sequence of Partial Sums: $S_n = a_1 + a_2 + \dots + a_n$.

• Converge/Diverge: A sequence or series converges if it approaches a finite limit; otherwise, it diverges.

Skills

• Write out terms of a sequence or series.

• Determine convergence and limits.

• Define sequences recursively or explicitly.

• Find limits of partial sums.

Example: Learning Curve Model

• Given $\frac{dP}{dt} = k(M - P)$, where $M$ is the maximum performance and $k > 0$.

• This models learning as the rate of improvement proportional to the remaining potential.

• Solution: $P(t) = M - (M - P_0) e^{-kt}$, where $P_0$ is initial performance.

Section 10.2 - Sequences

Definitions and Properties

• Increasing/Decreasing: $a_{n+1} > a_n$ (increasing), $a_{n+1} < a_n$ (decreasing).

• Nondecreasing/Nonincreasing: $a_{n+1} \geq a_n$ (nondecreasing), $a_{n+1} \leq a_n$ (nonincreasing).

• Monotonic: Sequence is either always increasing or always decreasing.

• Bounded Above/Below: There exists a number $M$ such that $a_n \leq M$ (above), $a_n \geq m$ (below).

• Bounded: Both above and below.

• Geometric Sequence: $a_n = ar^{n-1}$ for some $a$ and $r$.

Skills and Examples

• Find limits using algebraic or squeeze theorem methods.

• Compare growth rates of sequences.

• Examples:

  • $\lim_{n \to \infty} e^{-n} 2 \sin(e^{-n}) = 0$ (since $e^{-n} \to 0$ and $\sin(e^{-n}) \to 0$)

  • Examples of sequences with given properties:

    • Nondecreasing: $a_n = n$

    • Increasing: $a_n = 2^n$

    • Decreasing and converges: $a_n = 1/n$

    • Nonincreasing and diverges: $a_n = -n$

    • Bounded above and below but does not converge: $a_n = (-1)^n$

    • Bounded above and increasing: $a_n = 1 - 1/n$

Section 10.3 - Infinite Series

Definitions

• Series: The sum $\sum_{n=1}^\infty a_n$.

• Sequence of Partial Sums: $S_n = a_1 + a_2 + \dots + a_n$.

• Convergent Series: If $\lim_{n \to \infty} S_n$ exists and is finite.

• Divergent Series: If the limit does not exist or is infinite.

• Geometric Series: $\sum_{n=0}^\infty ar^n$ converges if $|r| < 1$.

• Telescoping Series: Series where many terms cancel, simplifying the sum.

Skills

• Evaluate geometric sums: $S = \frac{a}{1 - r}$ for $|r| < 1$.

• Evaluate telescoping series by writing out terms and observing cancellation.

• Understand convergence/divergence properties.

Examples

• Sierpinski Triangle: Area after $n$ iterations: $A_n = 1 - \sum_{k=1}^n \left(\frac{1}{4}\right)^k$ (assuming initial area 1). As $n \to \infty$, area approaches $0$.

• Telescoping Series: $\sum_{k=2}^\infty \left(\frac{1}{k-1} - \frac{1}{k+1}\right)$ can be summed by writing out terms and observing cancellation.

Table: Examples of Sequences with Given Properties

Property	Example Sequence
Nondecreasing	$a_n = n$
Increasing	$a_n = 2^n$
Decreasing and converges	$a_n = 1/n$
Nonincreasing and diverges	$a_n = -n$
Bounded above and below, does not converge	$a_n = (-1)^n$
Bounded above and increasing	$a_n = 1 - 1/n$

Additional info:

• Some examples and explanations have been expanded for clarity and completeness.

• Formulas and solution methods are provided in standard calculus notation.