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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Exact Equations with Integrating Factor | youtube icon Topic Playlist

Sometimes a differential equation is not exact but can be made exact by multiplying by an integrating factor, usually called $\mu(x,y)$. Unlike the method of integrating factor in a first order linear ODEs, a major difficulty in finding an integrating factor to an inexact equations is solving an even more difficult partial differential equation. However, it's easier if we can find an integrating factor that is only a function of $x$ or only a function of $y$, but not both.

Calculating the Integrating Factor for Inexact Equations | youtube icon Explanation Playlist

Let's assume the following differential equation is not an exact equation.

$\displaystyle M(x,y) + N(x,y)y' = 0 \qquad\qquad \frac{\partial }{\partial x}N \neq \frac{\partial }{\partial y}M \qquad$ since we said the equation is not exact.

Multiply by an unknown function $\mu(x,y)$ to make the equation exact.

\begin{equation} \mu(x,y)M(x,y) + \mu(x,y)N(x,y)y' = 0 \end{equation}

For the equation to be exact, the same test must work with the partial derivatives of the $y'$ term and the term without $y'$.

\begin{equation} \frac{\partial }{\partial x}\Big(\mu(x,y)N(x,y)\Big) = \frac{\partial }{\partial y}\Big(\mu(x,y)M(x,y)\Big) \end{equation}

Using the product rule with the partial derivatives creates an even more difficult to solve partial differential equation.

\begin{equation} \left(\frac{\partial }{\partial x}\mu(x,y)\right)N(x,y) + \mu(x,y)\left(\frac{\partial }{\partial x}N(x,y)\right) = \left(\frac{\partial }{\partial y}\mu(x,y)\right)M(x,y) + \mu(x,y)\left(\frac{\partial }{\partial y}M(x,y)\right) \end{equation} \begin{equation} \frac{\partial \mu}{\partial x}N + \mu\frac{\partial N}{\partial x} = \frac{\partial \mu}{\partial y}M + \mu\frac{\partial M}{\partial y} \end{equation}

This is no easy way to find a solution to this partial differential equation, but sometimes the integrating factor is only a function of one variable instead of both variables. The equation become much simpler if we try $\mu(x,y)=\mu(x)$ as just a function of $x$ because $\displaystyle \frac{\partial \mu}{\partial y} = 0$ and solving for $\mu(x)$ is a differential equation of one variable instead of two variables. Rearranging gives a nice equation for $\mu$ and $\dfrac{\partial \mu}{\partial x} = \mu'(x)$.

\begin{equation} \mu(x,y)=\mu(x) \Rightarrow \frac{\partial \mu}{\partial y} = 0 \qquad\qquad \frac{\partial \mu}{\partial x}N + \mu\frac{\partial N}{\partial x} = 0M + \mu\frac{\partial M}{\partial y} \end{equation}

$\displaystyle \frac{\mu'(x)}{\mu(x)} = \frac{\dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}}{N} \qquad$ which must simplify to a function of $x$.

There's a similar equation when assuming $\mu(x,y)=\mu(y)$ with $\displaystyle \frac{\partial \mu}{\partial x} = 0$ and $\dfrac{\partial \mu}{\partial y} = \mu'(y).$

$\displaystyle \frac{\mu'(y)}{\mu(y)} = \frac{\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}}{M} \qquad$ which must simplify to a function of $y$.

However, sometimes there is no integrating factor that is function of just one of the variables, in which case finding the integrating factor becomes difficult to impossible.

Example 1 | youtube icon Solution Video

Find an integrating factor for the inexact equation and solve the resulting exact equation.

\begin{equation} 3y + \frac{y^{2}}{x} + (x+y)y' = 0 \end{equation}

Example 2 | youtube icon Solution Video

Find an integrating factor for the inexact equation and solve the resulting exact equation.

\begin{equation} 2xy^{3} + (3x^{2}y^{2} + x^{2}y^{3} + 1)y' = 0 \end{equation}

Example 3 | youtube icon Solution Video

Determine $a$ and $b$ in $\mu(x,y)=x^{a}y^{b}$ for an integrating factor.

\begin{equation} -y + (x+x^{6})y' = 0 \qquad\qquad \mu(x,y) = x^{a}y^{b} \end{equation}