Find the general solution of D3 2D2 8Dy 0SolutionGiven th

Find the general solution of (D^3 + 2D^2 - 8D)y = 0

Solution

Given that

(D³ + 2D² - 8D)y = 0

The auxialary equation is ,

r³ + 2r² - 8r = 0

r(r² + 2r - 8) = 0

r = 0 , (r² + 2r - 8) = 0

r² + 4r - 2r - 8 = 0

r(r + 4) -2(r + 4) = 0

(r + 4) (r -2) = 0

r + 4 = 0 , r - 2 = 0

r = -4 , r = 2

Let r₁ = 0 , r₂ = -4 , r₃ = 2

If the roots are real and distinct then the general solution is ,

y(x) = c₁e^r₁x + c₂e^r₂x + c₃e^r₃x

   y (x) =  c₁e^0.x + c₂e^-4.x + c₃e^2.x

   y(x) = c₁e⁰ +  c₂e^-4x + c₃e^2x

   y(x) = c₁ +  c₂e^-4x + c₃e^2x [ Since , e⁰ = 1 ]

Therefore,

The general solution is ,    y(x) = c₁ +  c₂e^-4x + c₃e^2x