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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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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asked 2020-10-28

Solve the linear equations by considering y as a function of x, that is, \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{y}\right\rbrace}={\left\lbrace{y}\right\rbrace}{\left\lbrace{\left({\left\lbrace{x}\right\rbrace}\right)}\right\rbrace}.{\left\lbrace{x}\right\rbrace}{\left\lbrace{y}\right\rbrace}'+{\left\lbrace{\left({\left\lbrace{1}\right\rbrace}+{\left\lbrace{x}\right\rbrace}\right)}\right\rbrace}{\left\lbrace{y}\right\rbrace}={\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{x}\right\rbrace}\right\rbrace}}}{\sin{{\left\lbrace{\left\lbrace{2}\right\rbrace}\right\rbrace}}}{\left\lbrace{x}\right\rbrace}\)

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Answer 1

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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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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