Question

Linear equations of second order with constant coefficients. Find all solutions on displaystyle{left(-infty,+inftyright)}.{y}text{}+{4}{y}={0}

Second order linear equations
ANSWERED
asked 2021-01-05
Linear equations of second order with constant coefficients. Find all solutions on \(\displaystyle{\left(-\infty,+\infty\right)}.{y}\text{}+{4}{y}={0}\)

Answers (1)

2021-01-06

The characteristic equation is given by
\(r^{2}+4=0\)
and has the roots \(r_1=2i\) and \(r_2= -2i\)
The general solution is given by
\(y={c}_{{1}} \cos{{2}}{x}+{c}_{{2}} \sin{{2}}{x}\), 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.

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