which of the following answers is a solution to the differential equation below

Question

y''' - 6y'' + 11y' - 6y = 0

A. y = e^-x

B. y = e^4x

C. y = e^-3x

D. y = e^3x

Answer 1

Let y=e^rx and sub into the differential equation:

(e^rx)'''-6(e^rx)''+11(e^rx)'-6e^rx=0

We then calculate the derivatives.

r³e^rx-6r²e^rx+11re^rx-6e^rx=0

Factor out e^rx

e^rx(r³-6r²+11r-6)=0

Since e^rx is never zero we must have that

r³-6r²+11r-6=0

Factoring this we get

(r-1)(r-2)(r-3)=0

So that we have solutions

e^x,e^2x, and e^3x

And one of these solution e^3x is our answer d)!

Answer 2

Characteristic equation for this equation: r³ - 6r² + 11r - 6 = 0, r³ - r² - 5r² + 5r + 6r - 6 = 0,

(r - 1)(r² - 5r + 6) = 0, (r - 1)(r - 2)(r - 3) = 0, r = 1, r = 2 and r = 3.

So, y = e^3x is solution of this ODE.

Answer is D.