7th Degree Differential Homogeneous Operator

Question

Question asks:

$1, 1 - i, i, i$ are the roots of $L ( r ) = 0$ (characteristic equation) where $L(D)$ is a $7^{\text {th}}$ ( seventh) order linear, homogeneous differential operator with constant coefficients.

Find the differential equation $L (D) y = 0$ and its general solution.

Which way should i follow to solve this problem ?

Answer 1

$$ 1 \implies (r-1)$$ $$i \implies (r^2+1)$$ Again $i$: $$i \implies (r^2+1)^2$$ $$(1-i) \implies (r-(1-i))(r-(1+i))$$ So that you have: $$P(r)=(r-1)(r^2+1)^2(r-(1-i))(r-(1+i))$$ $$P(r)=(r-1)(r^2+1)^2((r-1)^2+1)$$ You can easily deduce the differential equation from the characteristic polynomial: $$((D-1)(D^2+1)^2((D-1)^2+1))y=0$$ Where $D=\dfrac {d}{dx}$

Answer 2

Regarding the general solution, you just need to add up the solutions coming from each factor in the characteristic polynomial... $$ y = c_1 e^x + \underbrace{(c_2x+c_3)\sin x + (c_4x+c_5)\cos x} + \underbrace{(c_6 \sin x +c_7 \sin x)e^x}. $$