Find a general solution to :y'' + 2y' + 4y = 0.

First we will rewrite into auxiliary equation
form.

==> r^2 + 2r + 4 = 0

Now
we will calculatet the roots.

==> r1= [ -2+ sqrt(4--16) / 2 =
-1+sqrt3*i

==> r2=
-1-sqrt3*i

Since the roots are not real, then we know
that:

==> a = -1 ==> B=
sqrt3

Then the solution is given by
:

y(x)= c1e^-x* cos(sqrt3 x) + c2*e^-x *
sin(sqrt3*x)

Then the general solution is given by
:

==> y(x) = e^-x [ c1*cos(sqrt3*x)
+ c2*sin(sqrt3* x)].