Printable View
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
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$.
