MHB Converting a Second-Order IVP into a System of Equations: Can Substitution Help?

  • Thread starter Thread starter karush
  • Start date Start date
  • Tags Tags
    Ivp Second order
karush
Gold Member
MHB
Messages
3,240
Reaction score
5
source

Change the second-order IVP into a system of equations
$y''+y'-2y=0 \quad y(0)= 2\quad y'(0)=0$

let $u=y'$

ok I stuck on this substitution stuff
 
Physics news on Phys.org
Since y'= u, y''= u' so y''+ y'- 2y= u'+ u- 2y= 0 so u'= 2y-u.
 
There is the following linear Volterra equation of the second kind $$ y(x)+\int_{0}^{x} K(x-s) y(s)\,{\rm d}s = 1 $$ with kernel $$ K(x-s) = 1 - 4 \sum_{n=1}^{\infty} \dfrac{1}{\lambda_n^2} e^{-\beta \lambda_n^2 (x-s)} $$ where $y(0)=1$, $\beta>0$ and $\lambda_n$ is the $n$-th positive root of the equation $J_0(x)=0$ (here $n$ is a natural number that numbers these positive roots in the order of increasing their values), $J_0(x)$ is the Bessel function of the first kind of zero order. I...
Are there any good visualization tutorials, written or video, that show graphically how separation of variables works? I particularly have the time-independent Schrodinger Equation in mind. There are hundreds of demonstrations out there which essentially distill to copies of one another. However I am trying to visualize in my mind how this process looks graphically - for example plotting t on one axis and x on the other for f(x,t). I have seen other good visual representations of...

Similar threads

Back
Top