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Curiosity on second order ODE - salvomic - 02-06-2015 10:02 AM
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
RE: Curiosity on second order ODE - parisse - 02-06-2015 11:44 AM
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.
RE: Curiosity on second order ODE - salvomic - 02-06-2015 12:07 PM
(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
RE: Curiosity on second order ODE - salvomic - 05-13-2015 08:13 PM
solved now with firmware 7820!
Answer: G_0*e^(3x)+G_1*e^(-2x)