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Given that $a, b, c$ are positive constants and $y(x)$ is a solution to the differential equation $ay''+by'+cy=0$, show that $\lim\limits_{x \to \infty} y(x) = 0$.

I've been able to determine that the general solution $y(x)=c_1e^{r_1x} + c_2e^{r_2x}$ should have $r_1$ and $r_2$ be negative (I found this by playing around with the numbers in GeoGebra). As for the proper solution though, I'm at a loss.

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The characteristic equation is $$az^2+bz+c=0.$$ Assuming real roots, they must be negative, otherwise the expression would be positive.

Assuming complex conjugate roots, their real part is $-b/2a$, a negative number.

In all cases, the solution decays exponentially to $0$.

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