Curiosity on second order ODE
02-06-2015, 10:02 AM (This post was last modified: 02-06-2015 10:28 AM by salvomic.)
Post: #1
 salvomic Senior Member Posts: 1,394 Joined: Jan 2015
Curiosity on second order ODE
hi,
The second order ordinary differential equation y''(x)-y'(x)-6 y(x) = 0 is solved into y(x) = c_1 e^(-2 x)+c_2 e^(3 x).
Why HP Prime give a factor of $$-\frac{1}{5}$$ added?
It give $$-\frac{1}{5}e^{-2x}(-3G_0+G_1)+\frac{1}{5}e^{3x}(2G_0+G_1)$$
With substitution of -3G_0+G1=c_1 and 2G_0+G1=c_2 we get the solution, but multiplied by 1/5.

Therefore, if the ODE isn't homogeneous (i.e. like y''(x)-y'(x)-6 y(x) = t*e^(-2t) we get a long expression with -1/15 ... 1/10 ... (try by yourself) instead of a "simply" expression (see here in Wolfamalpha)

I would like to simplify it a bit
Definitely, I would have a method to collect various constants (G_0, G_1...) to have a more compact format, like we do solving manually the equation. Maybe in the future Prime will do the job for us...

Thank you for reply and patience.

Salvo

∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C - DM42, DM41X - WP34s Prime Soft. Lib
02-06-2015, 11:44 AM
Post: #2
 parisse Senior Member Posts: 1,186 Joined: Dec 2013
RE: Curiosity on second order ODE
The current solution is expressed in terms of the initial value at 0.
At the request of a few Prime and Geogebra users, I have implemented a new method for general solutions of linear equations of order 2 with constant coeffs (also order 3 for generic case), it is already available in Xcas unstable versions.
02-06-2015, 12:07 PM
Post: #3
 salvomic Senior Member Posts: 1,394 Joined: Jan 2015
RE: Curiosity on second order ODE
(02-06-2015 11:44 AM)parisse Wrote:  The current solution is expressed in terms of the initial value at 0.
At the request of a few Prime and Geogebra users, I have implemented a new method for general solutions of linear equations of order 2 with constant coeffs (also order 3 for generic case), it is already available in Xcas unstable versions.

thank you!
Now it's clear.

I'm looking forward to see this and others improvements also in the Prime soon

∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C - DM42, DM41X - WP34s Prime Soft. Lib
05-13-2015, 08:13 PM
Post: #4
 salvomic Senior Member Posts: 1,394 Joined: Jan 2015
RE: Curiosity on second order ODE
solved now with firmware 7820!