What is the solution for (D^4+8D^2+16)y=0 ?

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

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(D^4+8D^2+16)y=0

(D^{4} + 8 D^{2} + 16) y = 0

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

Solution steps

(D^{4} + 8 D^{2} + 16) y = 0

Solve linear ODE:

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

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

- 6 y^{^{''}} + 54 y^{^{'}} = 0, y^{^{'}} (- 1) = 1, y (2) = 1

y^{^{''}} - 5 y^{^{'}} + 12 y = 0

25 y^{^{''}} + 70 y^{^{'}} + 49 y = 0

y^{^{''}} - y^{^{'}} - 20 y = 0, y (0) = 1, y^{^{'}} (0) = - 1

6 f^{^{''}} (t) = 3 f^{^{''}} (t) - 5 f (t)

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