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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Bernoulli Equations and Substitutions | youtube icon Topic Playlist

In general, no one alive or dead can find the solution to a nonlinear differential equation. The reason so much time is spent studying linear differential equations is not because nonlinear differential equations are less important, it's that nonlinear differential equations are nearly impossible to solve. Sometimes though, specific forms of nonlinear differential equations can be solved. This is so rare that the differential equations will be named. One of those types of equations is Bernoulli equations.

Bernoulli Substitution | youtube icon Explanation Video

A Bernoulli differential equation has the following form,

\begin{equation} y' + p(x)y = g(x)y^{n} \end{equation}

where $n$ is any real number except 0 or 1 because $n=0$ means the RHS is just $g(x)y^{0}=g(x)$ and $n=1$ is also linear by combining the $p(x)y$ and $g(x)y$ into one $(p(x)-g(x))y$ term.

Using the substitution $u = y^{1-n}$ results in a linear differential equation.

\begin{equation} u = y^{1-n} \qquad u' = (1-n)y^{-n}y' \end{equation} \begin{equation} y' + p(x)y = g(x)y^{n} \qquad u' + (1-n)p(x)u = (1-n)g(x) \end{equation}

Bernoulli Example 1 | youtube icon Solution Video

Find the general solution to the Bernoulli differential equation.

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

Bernoulli Example 2 | youtube icon Solution Video

Find the general solution to the Bernoulli differential equation.

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

Bernoulli Example 3 | youtube icon Solution Video

Find the general solution to the Bernoulli differential equation.

\begin{equation} y' + y = \sqrt{y} \end{equation}

Bernoulli Example 4 | youtube icon Solution Video

Find the general solution to the Bernoulli differential equation.

\begin{equation} y' + y = e^{x} y^{\frac{3}{7}} \end{equation}

Another common substitution problem is $u = \dfrac{y}{x}$ but the problems always seem a little made up.

$u = \dfrac{y}{x}$ Substitution | youtube icon Explanation Video

\begin{equation} u = \frac{y}{x} \qquad xu = y \qquad xu' + u = y' \end{equation} \begin{equation} y' = F\left(\frac{y}{x}\right) \qquad xu' + u = F(u) \end{equation}

$u = \dfrac{y}{x}$ Example 1 | youtube icon Solution Video

Find the general solution to the following differential equation.

\begin{equation} x^{2}y' = y^{2} + xy \end{equation}

$u = \dfrac{y}{x}$ Example 2 | youtube icon Solution Video

Find the general solution to the following differential equation.

\begin{equation} x^{2}y' = y^{2} + xy + x^{2} \end{equation}

Lastly, some differential equations were intentionally made for a particular substitution.

Contrived Example 1 | youtube icon Solution Video

Find the general solution to the following differential equation.

\begin{equation} yy' + x = \sqrt{x^{2}+y^{2}} \end{equation}

Contrived Example 2 | youtube icon Solution Video

Find the general solution to the following differential equation.

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