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Given $x_1(t)$ and $x_2(t)$ two solutions of a homogeneous differential equation, what can we say about the division of those?

Is $x_1(t)/x_2(t) $ a solution of the homogeneous equation too?

I have to calculate the general solution of a non-homogeneous differential equation of order 2 ($tx''+(2-3t)x'-3x=2t^3-15t^4)$ and the only information I have is that the division of two solutions of the homogeneous differential equation is $Tan(t)$. So, how can I start?

Thank you for your time!

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  • $\begingroup$ Is this all of the information you are provided with? What about the domain and the order of the equation? $\endgroup$ – user1337 Oct 28 '13 at 14:29
  • $\begingroup$ Isn't the first term in the ODE like $t^2x''$? $\endgroup$ – mrs Oct 28 '13 at 17:01
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In general the answer is no. Note that for $x_2(t) \equiv 0$ for example the division $\frac{x_1}{x_2}$ isn't even defined.

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  • $\begingroup$ But in this case, as the division is tan(t), can we assume it? $\endgroup$ – Blanca Oct 28 '13 at 14:26

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