The nth order linear differential equation with the constant coefficients when time goes to an infinity
By Michael Henderson •
We have the nth order linear differential equation with the constant coefficients such that $a_ny^n + a_{n-1}y^{n-1}+...+ a_1y' + a_0y= 0$. What would be the properties of the solution if the time would go to an infinity?
$\endgroup$1 Answer
$\begingroup$The properties of the sulution depent on $a_n,...,a_1,a_0$.
Examples: 1. $y''+y=0$. This equation has the general solution $y(x)=c_1 \sin x +c_2 \cos x$. If $c_1 \ne 0$ or $c_2 \ne 0$ then $\lim_{x \to \infty}y(x)$ doe not exist.
- $y''+2y'+y=0$ has the general solution $y(x)=c_1 e^{-x} +c_2 x e^{-x}$. We have $\lim_{x \to \infty}y(x)=0$
