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# Linear equations of second order with constant coefficients. Find all solutions on displaystyle{left(-infty,+inftyright)}.{y}text{}-{2}{y}'+{5}{y}={0} # 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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Second order linear equations asked 2021-02-21
Linear equations of second order with constant coefficients.
Find all solutions on
$$\displaystyle{\left(-\infty,+\infty\right)}.{y}\text{}-{2}{y}'+{5}{y}={0}$$

## Answers (1) 2021-02-22
The characteristic equation is given by
$$r^{2}+2r+5=0$$
and has the roots r_1=1+2i and r_2=1-2i.
The general solution is given by
$${y}={e}^{x}{\left({c}_{{1}} \cos{{2}}{x}+{c}_{{2}} \sin{{2}}{x}\right)}$$, 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.

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