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

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.