# Question #656be

Jul 13, 2016

$y = {C}_{1} {e}^{{\lambda}_{1} x} + {C}_{2} {e}^{{\lambda}_{2} x}$ with
${\lambda}_{1} = - \frac{1}{2} \left(2 + \sqrt{3}\right)$ and
${\lambda}_{2} = - \frac{1}{2} \left(2 - \sqrt{3}\right)$

#### Explanation:

This is a linear homogeneous differential equation with constant coefficients. For this equation, the solution has the structure

$y = C {e}^{\lambda x}$

substituting we have

$\left(2 {\lambda}^{2} + 4 \lambda + \frac{1}{2}\right) C {e}^{\lambda x} = 0$

but $C {e}^{\lambda x} \ne 0$ so the feasible $\lambda$'s are the solutions of

$2 {\lambda}^{2} + 4 \lambda + \frac{1}{2} = 0$

which have as solutions

${\lambda}_{1} = - \frac{1}{2} \left(2 + \sqrt{3}\right)$ and
${\lambda}_{2} = - \frac{1}{2} \left(2 - \sqrt{3}\right)$

then, the general solution is

$y = {C}_{1} {e}^{{\lambda}_{1} x} + {C}_{2} {e}^{{\lambda}_{2} x}$