Linear equations of second order with constant coefficients. Find all solutions on displaystyle{left(-infty,+inftyright)}.{y}text{}-{2}{y}'+{5}{y}={0}

Question

Linear equations of second order with constant coefficients.
Find all solutions on

(- \infty, + \infty) . y - 2 y^{'} + 5 y = 0

Answer 1

The characteristic equation is given by
$r^{2} + 2 r + 5 = 0$
and has the roots $r_{1} = 1 + 2 i$ and $r_{2} = 1 - 2 i .$
The general solution is given by
$y = e^{x} (c_{1} \cos 2 x + c_{2} \sin 2 x)$ , where $c_{1} and c_{2}$ are arbitracy constant.
This differential equation has infinitely many solutions since there is no initial conditions to determine a particular solution.

Linear equations of second order with constant coefficients. Find all solutions on displaystyle{left(-infty,+inftyright)}.{y}text{}-{2}{y}'+{5}{y}={0}

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