What is the solution for (D^5+8D^3+16D)y=0 ?

The solution for (D^5+8D^3+16D)y=0 is y=c_{1}+c_{2}cos(2x)+c_{3}sin(2x)+x(c_{4}cos(2x)+c_{5}sin(2x))

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(D^5+8D^3+16D)y=0

(D^{5} + 8 D^{3} + 16 D) y = 0

y = c_{1} + c_{2} cos (2 x) + c_{3} sin (2 x) + x (c_{4} cos (2 x) + c_{5} sin (2 x))

Solution steps

(D^{5} + 8 D^{3} + 16 D) y = 0

Solve linear ODE:

y = c_{1} + c_{2} cos (2 x) + c_{3} sin (2 x) + x (c_{4} cos (2 x) + c_{5} sin (2 x))

y = c_{1} + c_{2} cos (2 x) + c_{3} sin (2 x) + x (c_{4} cos (2 x) + c_{5} sin (2 x))

y^{^{''}} + y^{^{'}} - 8 y = 0

y^{^{''}} + 2 y^{^{'}} - 2 y = 0

y^{^{''}} - 3 y^{^{'}} - 10 y = 0

y^{^{''}} + 14 y^{^{'}} + 49 y = 0, y (- 4) = - 1, y^{^{'}} (- 4) = 5

a y^{^{''}} + b y^{^{'}} + cy = 0

What is the solution for (D^5+8D^3+16D)y=0 ?
The solution for (D^5+8D^3+16D)y=0 is y=c_{1}+c_{2}cos(2x)+c_{3}sin(2x)+x(c_{4}cos(2x)+c_{5}sin(2x))