(D^{3} - 14D + 8) y = 0

Step 1. Differential equation
Homogeneous DE with constant coefficient- auxiliary equations:
${\displaystyle (D^3-14D+8)y=0}$
Differential equation:
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Step 2. Solution
${\displaystyle D=m}$
${\displaystyle m^3-14m+8=0}$
${\displaystyle m=-4}$ is a factor,
${\displaystyle \Rightarrow m^3+4m^2-4m^2-16m+2m+8=0}$
${\displaystyle \Rightarrow m^2(n+4)-4m(m+4)+2(m+4)=0}$
${\displaystyle \Rightarrow (m+4)(m^2-4m+2)=0}$
${\displaystyle m=-4}$,
${\displaystyle m=\frac{4\pm \sqrt{16-8}}{2}}$
${\displaystyle m=\frac{4\pm 2\sqrt{2}}{2}}$
${\displaystyle m=2\pm \sqrt{2}}$
Step 3. Solution
The complementary function is:
${\displaystyle y_c=c_1{e}^{-4x}+e^{2x}(c_2\cos \sqrt{2x}+c_3\sin \sqrt{2x})}$
${\displaystyle c_1,c_2,c_3}$ are arbitrary constant.