Answer to Question #285011 in Differential Equations for godwynne

Question

y’’’’-5y’’’+6y’’+4y’-8y=0

Answer 1

Characteristic (auxiliary) equation

$$r^4-5r^3+6r^2+4r-8=0$$

$$(r+1)(r-2)^3=0$$

$$r_1=-1, r_2=r_3=r_4=2$$

The general solution of the given homogeneous differential equation is

$$t=c_1e^{-x}+c_2e^{2x}+c_3xe^{2x}+c_4x^2e^{2x}$$