Find the solution of the differential equation,
$\displaystyle \frac{d^4y}{dx^4}-4 \cdot \frac{d^3y}{dx^3}+8 \cdot \frac{d^2y}{dx^2}-8 \cdot \frac{dy}{dx}+4y=0$
Pl help to solve it. Thanks in adv..
The characteristic equation is $\displaystyle r^{4}-4r^{3}+8r^{2}-8r+4 = (r^{2}-2r+2)^{2} = 0 $
which has roots $\displaystyle 1 \pm i $, both of multiplicity 2
so the general solution is $\displaystyle y(x) = C_{1}e^{x} \cos(x) + C_{2} e^{x} \sin x + C_{3} e^{x}x \cos x + C_{4} e^{x}x \sin x $