All Mathematics

Overview

Welcome to your Civil Engineering Board Exam mathematics review! In this session, we'll break down essential calculus and differential equations concepts that often appear on board exams. You'll uncover how to recognize equation types, solve standard forms, and correctly integrate fundamental functions. By the end, you'll be able to approach these problems with confidence and clarity, equipped with proven strategies and a deeper conceptual understanding.

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Concept-by-Concept Deep Dive

1. Differential Equations: Types, Order, and Solutions

A differential equation involves derivatives of an unknown function. They're classified by order (the highest derivative present) and linearity (whether the unknown function and its derivatives appear to the first power and are not multiplied together).

Order of a Differential Equation

• The order is the degree of the highest derivative. For example, $\frac{d^3y}{dx^3} = 0$ is a third-order equation because the third derivative is the highest.
• To determine order, scan the equation for the highest derivative.

Linear vs Nonlinear Equations

• Linear: The unknown function and its derivatives appear linearly (no products or powers of $y$ or its derivatives).
• Nonlinear: The equation includes terms like $(y')^2$, $y \cdot y''$, etc.

Homogeneous vs Nonhomogeneous

• Homogeneous (in context): All terms involve the unknown function or its derivatives; no standalone functions of $x$ or constants.
• Nonhomogeneous: Includes terms free of the unknown function (e.g., $y'' + y = \sin x$).

General and Particular Solutions

• The general solution contains arbitrary constants (equal to the order of the equation).
• A particular solution results when initial or boundary conditions are applied.

Recipe for Solving First-Order Differential Equations

1. Identify the equation form ($\frac{dy}{dx} = f(x)$, separable, linear, exact, etc.).
2. For separable: Rearrange to get all $y$ terms with $dy$, all $x$ terms with $dx$, then integrate both sides.
3. For linear: Put in standard form $\frac{dy}{dx} + P(x)y = Q(x)$, find the integrating factor, and solve.

Common Misconceptions

• Confusing order with degree (degree is about powers, order is about derivatives).
• Not realizing when a general solution is required (missing constants).
• Misidentifying linearity due to complex-looking terms.

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2. Integrals of Standard Functions

Integration is the process of finding the antiderivative, or the original function whose derivative gives the integrand.

Basic Integrals

• Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
• Exponential Functions: $\int e^x dx = e^x + C$
• Trigonometric Functions:
  • $\int \sin x dx = -\cos x + C$
  • $\int \cos x dx = \sin x + C$
  • $\int \sec^2 x dx = \tan x + C$

Integration Steps

1. Identify the function's form (power, exponential, trig).
2. Apply the corresponding formula.
3. Always include the constant of integration, $C$.

Common Misconceptions

• Forgetting the constant of integration.
• Mixing up signs in trigonometric integrals.
• Misapplying the power rule to exponentials or trigonometric functions.

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3. Integrating Factor and Linear First-Order Equations

An integrating factor (IF) is a function used to solve linear first-order differential equations of the form $\frac{dy}{dx} + P(x)y = Q(x)$.

Calculating the Integrating Factor

• The integrating factor is $IF = e^{\int P(x) dx}$.
• Multiply both sides of the equation by the integrating factor to make the left side a product derivative.

Steps to Solve

1. Identify $P(x)$ in the standard form.
2. Compute the integrating factor.