Given $ay''+by'+cy=0$ and assuming that $a, b, c > 0$ show that all solutions approach $0$ as $t\rightarrow\infty$
I was able to begin by seperating the problem into three cases:
Case 1: Repeated roots of the characteristic equation
I was able to find the limit as $t\rightarrow\infty$ for the general solution to Case 1 using l'Hospital's Rule: Fairly straightforward.
Case 2: Imaginary roots of the characteristic equation
I was able to find the limit as $t\rightarrow\infty$ for the general solution to Case 2 using the squeeze theorem: Also fairly straightforward.
Case 3: Two distinct real roots of the characteristic equation
In this case the two roots are: $$r_1=\frac{-b+\sqrt{b^2-4ac}}{2a}$$ $$r_2=\frac{-b-\sqrt{b^2-4ac}}{2a}$$
and the general solution is: $$y=c_1e^{r_1t}+c_2e^{r_2t}$$
The only way I see that $\lim_{t\to\infty} y=0$ is if $r_1,r_2<0$. However if the only information I am given is that $a,b,c>0$ how can I prove that $r_1,r_2<0$?