Solutions of a second order differential linear and homogeneous equation with constant coefficients:

a y” + b y’ + c y = 0 (a ≠ 0)

Solve the characteristic equation: a r^{2} + b r + c = 0

with general solutions r_{1,2}=[(-b±√b^{2}-4ac )/2a].

1> if Δ=b^{2}-4ac > 0 then the solutions r_{1}=r_{1} are real and

the general solution of the differential equation is

y = c

_{1 }e^{r1x}+ c_{2 }e^{r2x}2> if Δ=b^{2}-4ac = 0 then the solutions are real and equal (r_{1}=r_{>2}) and the general solution of the differential equation is

y = c

_{1 }e^{r1x}+ c_{2}x e^{r1x}3> if Δ=b^{2}-4ac < 0 then the solutions are complex and the general solution of the differential equation is

y = c

_{1 }e^{kx}cos(lx) + c_{2 }e^{kx}sin(lx)where r_{1,2}=k±il=(-b/2a)±i[√b^{2}-4ac /2a]