Curiosity on second order ODE

[salvomic]

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

[parisse]

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.

[salvomic]

(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

[salvomic]

solved now with firmware 7820!

Answer: G_0*e^(3x)+G_1*e^(-2x)