Answer to Question #294036 in Differential Equations for Kej

Question

Find the solution of

(D^4+2D^3-6D^2-16D-8)y=0

Answer 1

Solution:

$(D^4+2D^3-6D^2-16D-8)y=0$

$\Rightarrow D^4+2D^3-6D^2-16D-8=0 \\\Rightarrow \left(D+2\right)^2\left(D^2-2D-2\right)=0 \\ \Rightarrow D+2=0,D+2=0\quad \mathrm{or}\quad \:D^2-2D-2=0 \\\Rightarrow D=-2,D=-2,\:D=1+\sqrt{3},\:D=1-\sqrt{3}$

Thus, solution will be:

$y=c_1e^{\left(1+\sqrt{3}\right)x}+c_2e^{\left(1-\sqrt{3}\right)x}+c_3e^{-2x}+c_4xe^{-2x}$