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