Table of Contents
Differential Equations
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Courses

  1. Linear Algebra
  2. Multivariable Calculus
  3. Differential Equations
  4. Miscellaneous Topics

Differential Equations

  1. Introduction
    1. Notation and Definitions
    2. Verifying Solutions
    3. Initial Values Problems
  2. First Order Differential Equations
    1. Direction Fields
    2. Equilibria, Stability, and Phase Lines
    3. Separable Equations
    4. Integrating Factor
    5. Bernoulli Equations and Substitutions
    6. Exact Equations
    7. Exact Equations with Integrating Factor
    8. Euler's Method
    9. Existence and Uniqueness
    10. Interval of Validity
    11. Applications
      1. Radioactive Decay and Population Growth
      2. Mixing Tank Problem
      3. Terminal Velocity
      4. Continuous Compound Interest
  3. Wrońskian
  4. Second Order Differential Equations
    1. $ay''+by'+cy = 0$
      1. Distinct Real Roots
      2. Repeated Real Root
      3. Imaginary Roots
    2. Spring-Mass-Damper
      1. Equivalent RLC Circuit
      2. Underdamped, Overdamped, Critically Damped
      3. Undamped Oscillations and Resonance
    3. Method of Undetermined Coefficients
    4. Reduction of Order
    5. Variation of Parameters
    6. Cauchy-Euler Equations
  5. Laplace Transform
    1. Lookup Table and WolframAlpha
  6. Systems of ODEs
    1. Simplest Case $\begin{bmatrix}x' \\ y'\end{bmatrix} = \begin{bmatrix}a & b \\ c & d \end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}$
      1. Poincaré Diagram
  7. Preview of Dynamical Systems
    1. Self-Study
    2. Rössler Attractor

Additional Resources

  1. WolframAlpha Examples
  2. Direction Field Plotter by Ariel Barton
  3. Phase Plane Plotter by Ariel Barton


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Table of Laplace Transforms and WolframAlpha | youtube icon Topic Playlist

This page contains a list of Laplace Transforms for common functions. Rather than computing the Laplace transform every time, looking up the transform in the table is much easier. The table can also be used to perform the inverse Laplace Transform. If you need the Laplace Transform for a function not included in the table, you can use WolframAlpha LaplaceTransform[ f(t), t, s] to compute it. Keep in mind that even a simple function can have a complicated Laplace Transform.

WolframAlpha Laplace Transform Example 1

The above link leads to WolframAlpha with the following input.

LaplaceTransform[ sqrt(t), t, s]

Part of the output is

\begin{equation} \frac{\sqrt{\pi}}{2 s^{3/2}} \end{equation}

WolframAlpha Laplace Transform Example 2

The above link leads to WolframAlpha with the following input.

LaplaceTransform[ HeavisideTheta[t-4]*e^(3t), t, s]

Part of the output is

\begin{equation} \frac{e^{12 - 4 s}}{s - 3} \end{equation}

WolframAlpha Laplace Transform Example 3

The above link leads to WolframAlpha with the following input.

LaplaceTransform[ DiracDelta[t-2]*e^(-3t), t, s]

Part of the output is

\begin{equation} e^{-2(s+3)} \end{equation}

WolframAlpha Inverse Laplace Transform Example 1

The above link leads to WolframAlpha with the following input.

InverseLaplaceTransform[ e^(-3s)*2/(s^2+4), s, t]

Part of the output is

\begin{equation} \theta(t - 3) \sin\big(2 (t - 3)\big) \end{equation}

Where $\theta(t)$ is the Heaviside function.

WolframAlpha Inverse Laplace Transform Example 2

The above link leads to WolframAlpha with the following input.

InverseLaplaceTransform[ 3*e^(-5s), s, t]

Part of the output is

\begin{equation} 3 \delta(t - 5) \end{equation}

Where $\delta(t)$ is the Dirac delta function.

Common Functions and their Laplace Transforms

$f(t) = \mathcal{L}^{-1}\{ F(s) \}$ $\mathcal{L}\{ f(t) \} = F(s)$
$cf(t)$ $cF(s)$
$1$ $\dfrac{1}{s}$
$e^{at}$ $\dfrac{1}{s-a}$
$t^{n} \text{ for } n=1,2,3,\ldots$ $\dfrac{n!}{s^{n+1}}$
$\sin(at)$ $\dfrac{a}{s^{2}+a^{2}}$
$\cos(at)$ $\dfrac{s}{s^{2}+a^{2}}$
$t\sin(at)$ $\dfrac{2as}{(s^{2}+a^{2})^{2}}$
$t\cos(at)$ $\dfrac{s^{2}-a^{2}}{(s^{2}+a^{2})^{2}}$
$e^{at}\sin(bt)$ $\dfrac{b}{(s-a)^{2}+b^{2}}$
$e^{at}\cos(bt)$ $\dfrac{s-a}{(s-a)^{2}+b^{2}}$
$t^{n}e^{at} \text{ for } n=1,2,3,\ldots$ $\dfrac{n!}{(s-a)^{n+1}}$
$H(t-c)$ $\dfrac{e^{-cs}}{s}$
$\delta(t-c)$ $e^{-cs}$
$H(t-c)f(t-c)$ $e^{-cs}F(s)$
$f'(t)$ $sF(s)-f(0)$
$f''(t)$ $s^{2}F(s)-sf(0)-f'(0)$
$f^{(n)}(t)$ $s^{n}F(s)-s^{n-1}f(0)-s^{n-2}f'(0)- \cdots -sf^{(n-2)}(0)-f^{(n-1)}(0)$