Results 1 to 2 of 2

Thread: Equi Potential Equations or Euler Type Equations

  1. #1
    Apr 2010

    Equi Potential Equations or Euler Type Equations

    Hello All

    I want to know about "Equi-Potential Equations" which are also known as "Euler Type Equations". I have less exposure to these equations. I also searched on net but didn't get a clear view of it. Can you explain or provide some references regarding these equations.

    Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Apr 2005
    The main thing you need to know about "Euler-type" equations is that there is a "one-to-one" correspondence between them and equations with constant coefficients.

    Just as you can "look for" solutions to equations with constant coefficients using $\displaystyle e^{rx}$ (even though solutions may not be exponential), so you can "look for" solutions to "Euler-type" equations using $\displaystyle x^r$.

    That's because the change of variable x= ln(t) will change an "Euler-type" equation in variable t to an equation with constant coefficients in variable t.

    For example, $\displaystyle at^2 \frac{d^2y}{dt^2}+ bt\frac{dy}{dt}+ cy= 0$ is an "Euler-type" equation in variable t. If we let x= ln(t), then $\displaystyle \frac{dy}{dt}= \frac{dy}{dx}\frac{dx}{dt}= \frac{1}{t} \frac{dy}{dx}$.

    Further $\displaystyle \frac{d^2y}{dt^2}= \frac{d}{dt}\left(\frac{1}{t} \frac{dy}{dx}\right)$$\displaystyle = \frac{1}{t}\frac{d}{dt}\left(\frac{1}{t}\frac{dy}{ dx}\right)= \frac{1}{x}\left(\frac{1}{t}\frac{d^2y}{dx}^2- \frac{1}{t^2}\frac{dy}{dx}\right)$.

    So $\displaystyle at^2\frac{d^2y}{dt^2}+ bt\frac{dy}{dt}+ cy$$\displaystyle = a\left(\frac{d^2y}{dx^2}- \frac{dy}{dx}\right)+ b\frac{dy}{dx}+ cy$$\displaystyle = a\frac{d^2y}{dx^2}+ (b- a)\frac{dy}{dx}+ cy= 0$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Solving equations, but what type
    Posted in the Algebra Forum
    Replies: 11
    Last Post: Sep 10th 2011, 01:03 AM
  2. Solving x = log(x) type equations.
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Sep 1st 2011, 03:46 PM
  3. type of partial differential equations?
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: Jun 22nd 2010, 04:50 PM
  4. How to handle this type of equations?
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: Jan 23rd 2010, 08:20 AM
  5. Howto type equations
    Posted in the LaTeX Help Forum
    Replies: 6
    Last Post: Nov 28th 2008, 05:35 PM

Search Tags

/mathhelpforum @mathhelpforum