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)].