You CAN Ace Differential Equations

 precalculus - even and odd functions infinite series basics of differential equations

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

free ideas to save on books

This page covers two areas related to Fourier Series. First, we present an introduction to Fourier Series, then we discuss how to solve differential equations using Fourier Series. If you are just learning about Fourier Series, you can go through the introduction and practice problems and skip the section related to solving differential equations.

 What is a Fourier Series?

The main idea of Fourier Series is that we want to build an infinite series, using the basic trig functions sine and cosine, that is equivalent to a more complicated function. The series can then be manipulated more easily than the original function.

Here is a great video to get you started. He explains why we need to build these functions, goes through an example and then explains the big picture.

### Dr Chris Tisdell - building functions [13min-52secs]

The solution to the practice problem at the very end of this video can be found in his free workbook found here.

video by Dr Chris Tisdell

 How to Calculate Fourier Series

As you saw in that video, there are some basic equations required to calculate the Fourier Series. To build a Fourier Series for a function $$f(t)$$ with period $$2L$$, it is required that $$f(t)$$ and it's derivative $$f'(t)$$ be piecewise continuous on the interval $$[-L,L]$$.

Fourier Series Equations

Fourier Series

$$\displaystyle{ f(t) = a_0 + \sum_{n=1}^{\infty}{ \left[ a_n \cos \frac{n \pi t}{L} + b_n \sin \frac{n \pi t}{L} \right] } }$$

constants

$$\displaystyle{ a_0 = \frac{1}{2L} \int_{-L}^{L}{f(t)~dt} }$$

$$\displaystyle{ a_n = \frac{1}{L} \int_{-L}^{L}{f(t)\cos \frac{n\pi t}{L} ~dt} }$$

$$\displaystyle{ b_n = \frac{1}{L} \int_{-L}^{L}{f(t)\sin \frac{n\pi t}{L} ~dt} }$$

Knowing if the original $$f(t)$$ is either even or odd can help us a lot when finding the Fourier Series. Of course, we do not require that $$f(t)$$ be even or odd, but you remember from precalculus that cosine is an even function and sine is odd. So, for even functions $$b_n=0$$ and for odd functions $$a_n=0$$.

Instructions - - Unless otherwise instructed, find the Fourier Series for these functions.

$$\displaystyle{ f(t) = \left\{ \begin{array}{rr} -1 & -\pi < t < 0 \\ 0 & t = 0, \pm \pi \\ 1 & 0 < t < \pi \end{array} \right. }$$
$$f(t) = f(t+2\pi)$$ for all $$t$$

Problem Statement

$$\displaystyle{ f(t) = \left\{ \begin{array}{rr} -1 & -\pi < t < 0 \\ 0 & t = 0, \pm \pi \\ 1 & 0 < t < \pi \end{array} \right. }$$
$$f(t) = f(t+2\pi)$$ for all $$t$$

Solution

### 1296 solution video

video by Dr Chris Tisdell

$$\displaystyle{f(x) = \left\{ \begin{array}{rr} 0 & -1 \leq x \leq 0 \\ 1 & 0 < x < 1 \end{array} \right. }$$
with period $$2$$

Problem Statement

$$\displaystyle{f(x) = \left\{ \begin{array}{rr} 0 & -1 \leq x \leq 0 \\ 1 & 0 < x < 1 \end{array} \right. }$$
with period $$2$$

Solution

### 1297 solution video

video by PatrickJMT

$$\displaystyle{ f(x) = \left\{ \begin{array}{rr} -3 & -1 < x < 0 \\ 3 & 0 < x < 1 \end{array} \right. }$$
$$f(x) = f(x+2)$$

Problem Statement

$$\displaystyle{ f(x) = \left\{ \begin{array}{rr} -3 & -1 < x < 0 \\ 3 & 0 < x < 1 \end{array} \right. }$$
$$f(x) = f(x+2)$$

Solution

### 1298 solution video

video by Dr Chris Tisdell

 Solving Differential Equations

These videos show how to use Fourier Series to solve differential equations.

video by MIT OCW

video by MIT OCW

video by MIT OCW