Answer to Question #340401 in Differential Equations for Mqondisi

Question

(D^4 − 18D^3 + 119D^2 − 342D + 360)y = 0

Answer 1

Solution

For given homogeneous equation the characteristic equation is

λ⁴ – 18 λ³ + 119 λ²– 342 λ + 360 = 0

λ⁴ – 3 λ³ – 15 λ³ + 45 λ² + 74 λ² – 222 λ – 120 λ + 360 = 0

(λ – 3)( λ³ – 15 λ² + 74 λ – 120 ) = 0

(λ – 3)( λ³ – 4 λ² – 11 λ² + 44 λ + 30 λ – 120 ) = 0

(λ – 3) (λ – 4) ( λ² – 11 λ + 30 ) = 0

(λ – 3) (λ – 4) (λ – 5) (λ – 6) = 0

So solutions of this equation are λ₁ = 3, λ₂ = 4, λ₃ = 5, λ₄ = 6.

Therefore the general solution of the given differential equation is

y(x) = Ae^3x + Be^4x + Ce^5x + De^6x , where A, B, C, D are arbitrary constants. 

Answer

y(x) = Ae^3x + Be^4x + Ce^5x + De^6x