Question

Find the Solution of

(D^3-3D^2-3D+1)y=0

Answer 1

Solution

It is a linear homogeneous equation. The characteristic equation is

λ³ - 3λ² - 3λ + 1=0  =>  λ³ + 1 - 3 λ (λ + 1) = 0   =>   (λ + 1) (λ² - λ + 1) - 3 λ (λ + 1) = 0   =>  

(λ + 1) (λ² - 4λ + 1) = 0   

So there are three roots of the characteristic equation

$\lambda_1=-1,{\ \lambda}_2=2-\sqrt3,\ \lambda_3=2+\sqrt3$

Therefore the solution of given equation is

$y(x)=C_1e^{-x}+C_2e^{(2-\sqrt3)x}+C_3e^{(2+\sqrt3)x}$

where C₁, C₂, C₃ are arbitrary constants.