Question

Homogenous Linear Differential Equations

6. (D ^ 3 - 2D ^ 2 - 3D) * y = 0

5. (D ^ 3 - 3D ^ 2 + 4) * y = 0

7. (4D ^ 3 - 3D + 1) * y = 0

Answer 1

D³ - 2D² - 3D$\times$y = 0

(D-1)(D²-2D)=0

D=1, $\frac{2+\sqrt(4-8)}{2}$

y(x)= $\frac{1}{D^3-3D^2+4D-2}$ ex

y(x)= $\frac{1}{1-3+4-2}$ ex

y= $\frac{1}{1-3+4-2}$ x^ex

D³ - 3D² + 4)$\times$ y = 0

(D-1)(D²-3D+4)=0

D=1, $\frac{2+\sqrt(4-8)}{2}$

y(x)= $\frac{1}{D^3-3D^2+4D-2}$ ex

y(x)= $\frac{1}{1-3+4-2}$ ex

y= $\frac{1}{1-3+4-2}$ x^ex

^

(4D³ - 3D + 1)$\times$ y = 0

(D-1)(4D²-3D+1)=0

D=1, $\frac{3+\sqrt(4-3)}{2}$

y(x)= $\frac{1}{D^3-3D^2+4D-2}$ ex

y(x)= $\frac{1}{1-3+4-2}$ ex

y= $\frac{1}{1-3+4-2}$ x^ex