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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Method of Undetermined Coefficients | youtube icon Topic Playlist

This method can be used to solve differential equations of the form $ay''+by'+cy=g(x)$ where $g(x)$ is restricted to combinations of polynomials, exponentials, and function of sine and cosine. The method is really just a lookup table of guesses.

Simplest Case | youtube icon Explanation Video

In the simplest cases, the guess for the particular solution is a polynomial, an exponential, or a combination of sine and cosine.

ODE Initial $y_{p}(x)$ Guess
$ay''+by'+cy= 2x^{3}-1$ $Ax^{3}+Bx^{2}+Cx+D$
$ay''+by'+cy= 4e^{7x}$ $Ae^{7x}$
$ay''+by'+cy= 13\sin(2x)$ $A\cos(2x)+B\sin(2x)$

The capital letters $A,B,C,\ldots$ in the guesses are the undetermined coefficients, which is the name for the method. After guessing the form of the particular solution, the guess is plugged into the ODE and $A,B,C,\ldots$ are solved for to get the LHS to equal the RHS.

Example 1 | youtube icon Solution Video

Find the particular solution to the following differential equation.

\begin{equation} y'' - y' - 12y = 2x^{3}-1 \end{equation}

Example 2 | youtube icon Solution Video

Find the particular solution to the following differential equation.

\begin{equation} y'' - y' - 12y = 4e^{7x} \end{equation}

Example 3 | youtube icon Solution Video

Find the particular solution to the following differential equation.

\begin{equation} y'' - y' - 12y = 13\sin(2x) \end{equation}

When the Initial Guess is a Homogeneous Solution | youtube icon Explanation Video

Sometimes the RHS is a general solution. In that case, the normal guess is multiplied by $x$ until the $y_{p}(x)$ guess no longer contains a homogeneous solution.

ODE General Solution $y_{p}(x)$ Guess
$y''+y'= 2x^{3}-1$ $c_{1}e^{0x}+c_{2}e^{-x}=c_{1}+c_{2}e^{-x}$ $x(Ax^{3}+Bx^{2}+Cx+D)$
$y''-4y'-21y= 4e^{7x}$ $c_{1}e^{7x}+c_{2}e^{-3x}$ $Axe^{7x}$
$y''-14y'+49y= 4e^{7x}$ $c_{1}e^{7x}+c_{2}xe^{7x}$ $Ax^{2}e^{7x}$
$y''+4y= 13\sin(2x)$ $c_{1}\cos(2x)+c_{2}\sin(2x)$ $Ax\cos(2x)+Bx\sin(2x)$

It's always important to solve for the homogeneous solution first so you don't make a guess that won't work.

Example 4 | youtube icon Solution Video

Find the solution to the following differential equation.

\begin{equation} y''+y'= 2x^{3}-1 \end{equation}

Example 5 | youtube icon Solution Video

Find the solution to the following differential equation.

\begin{equation} y''-4y'-21y= 4e^{7x} \end{equation}

Example 6 | youtube icon Solution Video

Find the solution to the following differential equation.

\begin{equation} y''-14y'+49y= 4e^{7x} \end{equation}

Example 7 | youtube icon Solution Video

Find the solution to the following differential equation.

\begin{equation} y''+4y= 13\sin(2x) \end{equation}

Adding Guesses Together | youtube icon Explanation Video

If the RHS $g(x)$ is a sum of the functions that can be guessed for, $y_{p}(x)$ is just a sum of all those guesses.

ODE Initial $y_{p}(x)$ Guess
$ay''+by'+cy= 2x^{3}-1 + 4e^{7x} + 13\sin(2x)$ $Ax^{3}+Bx^{2}+Cx+D + Ae^{7x} + A\cos(2x)+B\sin(2x)$

Keep in mind that the particular solution can still require modification depending on the homogeneous solutions.

ODE and General Solution $y_{p}(x)$ Guess
$y''+y'= 2x^{3}-1 + 4e^{7x} + 13\sin(2x)$
$y_{h}(x) = c_{1}e^{0x}+c_{2}e^{-x}=c_{1}+c_{2}e^{-x}$		 $x(Ax^{3}+Bx^{2}+Cx+D) + Ee^{7x} + F\cos(2x)+G\sin(2x)$
$y''-4y'-21y= 2x^{3}-1 + 4e^{7x} + 13\sin(2x)$
$y_{h}(x) = c_{1}e^{7x}+c_{2}e^{-3x}$		 $Ax^{3}+Bx^{2}+Cx+D + Exe^{7x} + F\cos(2x)+G\sin(2x)$
$y''-14y'+49y= 2x^{3}-1 + 4e^{7x} + 13\sin(2x)$
$y_{h}(x) = c_{1}e^{7x}+c_{2}xe^{7x}$		 $Ax^{3}+Bx^{2}+Cx+D + Ex^{2}e^{7x} + F\cos(2x)+G\sin(2x)$
$y''+4y= 2x^{3}-1 + 4e^{7x} + 13\sin(2x)$
$y_{h}(x) = c_{1}\cos(2x)+c_{2}\sin(2x)$		 $Ax^{3}+Bx^{2}+Cx+D + Ee^{7x} + Fx\cos(2x)+Gx\sin(2x)$

Multiplying Guesses Together | youtube icon Explanation Video

If the RHS $g(x)$ is a product of the functions that can be guessed for, $y_{p}(x)$ is just a product of all those guesses.

ODE Initial $y_{p}(x)$ Guess
$ay''+by'+cy= (x^{2}-4) e^{7x} \sin(2x)$ $(Ax^{2}+Bx+C) e^{7x} \big(D\cos(2x)+E\sin(2x)\big)$
$C$ could be replaced with 1.

You only have to multiply the guess by $x$ if, after multiplying everything out, one of the terms is a constant times a homogeneous solution.

ODE and General Solution $y_{p}(x)$ Guess
$y''-y'-12y = (x^{2}-4) e^{7x} \sin(2x)$
$y_{h}(x) = c_{1}e^{-3x}+c_{2}e^{4x}$		 $(Ax^{2}+Bx+C) e^{7x} \big(D\cos(2x)+E\sin(2x)\big)$
$y''+y' = (x^{2}-4) e^{7x}$
$y_{h}(x) = c_{1}+c_{2}e^{-x}$		 $(Ax^{2}+Bx+C) e^{7x}$
$y''+y' = (x^{2}-4) e^{7x} \sin(2x)$
$y_{h}(x) = c_{1}+c_{2}e^{-x}$		 $(Ax^{2}+Bx+C) e^{7x} \big(D\cos(2x)+E\sin(2x)\big)$
$y''-7y' = e^{7x} \sin(2x)$
$y_{h}(x) = c_{1}+c_{2}e^{7x}$		 $e^{7x} \big(A\cos(2x)+B\sin(2x)\big)$
$y''-7y' = (x^{2}-4) e^{7x}$
$y_{h}(x) = c_{1}+c_{2}e^{7x}$		 $x(Ax^{2}+Bx+C) e^{7x}$
$y''-7y' = (x^{2}-4) e^{7x} \sin(2x)$
$y_{h}(x) = c_{1}+c_{2}e^{7x}$		 $(Ax^{2}+Bx+C) e^{7x} \big(D\cos(2x)+E\sin(2x)\big)$
$y''+8y'+25y = (x^{2}-4) e^{-4x} \sin(2x)$
$y_{h}(x) = c_{1}e^{-4x}\cos(3x)+c_{2}e^{-4x}\sin(3x)$		 $(Ax^{2}+Bx+C) e^{-4x} \big(D\cos(2x)+E\sin(2x)\big)$
$y''+8y'+25y = (x^{2}-4) e^{-4x} \sin(3x)$
$y_{h}(x) = c_{1}e^{-4x}\cos(3x)+c_{2}e^{-4x}\sin(3x)$		 $x(Ax^{2}+Bx+C) e^{-4x} \big(D\cos(3x)+E\sin(3x)\big)$
$y''+8y'+25y = (x^{2}-4) e^{-4x}$
$y_{h}(x) = c_{1}e^{-4x}\cos(3x)+c_{2}e^{-4x}\sin(3x)$		 $(Ax^{2}+Bx+C) e^{-4x}$
$y''+8y'+25y = (x^{2}-4) \sin(3x)$
$y_{h}(x) = c_{1}e^{-4x}\cos(3x)+c_{2}e^{-4x}\sin(3x)$		 $(Ax^{2}+Bx+C) \big(D\cos(3x)+E\sin(3x)\big)$