# Division of two solutions of a homogeneous differential equation

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?

• Isn't the first term in the ODE like $t^2x''$? – mrs Oct 28 '13 at 17:01
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.