The characteristic equation of the eighthorder differential

The characteristic equation of the eighth-order differential equation y(8) + 3y(7) + 7y(6) - y(5) - 19y(4) - 23y(3) - 3y\" + 85y\' - 50y = 0 factors as (r + 2) (r - 1)3 (r - [-1 + 2i])2(r - (-1 - 2i])2. Find the general form of a solution to this differential equation.

Solution

Substituting e^rx in the given characteristic equation we get following expression

(r⁸ + 3r⁷ + 7r⁶ - r⁵ -19r⁴ -23r³ -3r² +85r -50) e^rx = 0

Therefore, r⁸ + 3r⁷ + 7r⁶ - r⁵ -19r⁴ -23r³ -3r² +85r -50 = 0

Factorization of which is given as

(r+2)(r-1)³(r-[-1+2i])²(r-[-1-2i])² = 0

Therefore, general solution of this equation would be

y = Ae^-2x + Be^x + Cxe^x + Dx²e^x + Ee^(-1+2i)x + Fe^(-1-2i)x + Gxe^(-1+2i)x + Hxe^(-1-2i)x