Which of the following functions are solutions of the differ

Which of the following functions are solutions of the differential equation y??6y?+9y=0 ?

Which of the following functions are solutions of the differential equation y\" - 6y\' + 9y = 0? A. y(x) = e^3x B. y(x) = 3x C. y(x) = e^-3x D. y(x) = xe^-3x E.y(x) = 0 F. y(x) = x^2 e^3x G. y(x) = xe^3x

Solution

Given that

y\'\' - 6y\' + 9y = 0

Auxialary equation is ,

r² - 6r + 9 = 0

r² - 3r - 3r + 9 = 0

r(r - 3) - 3(r - 3) = 0

(r - 3)(r - 3) = 0

r - 3 = 0 , r - 3 = 0

r = 3 , r = 3

If the roots are real and equal then the solution is ,

y(x) = c₁e^rx + c₂xe^rx

y(x) = c₁e^3x + c₂xe^3x

Therefore ,

y(x) = e^3x , y(x) = xe^3x are solutions of given differential equation

......Options A , G are correct answers