What is the solution for (D^4+18D^2+81)y=0 ?

The solution for (D^4+18D^2+81)y=0 is y=c_{1}cos(3x)+c_{2}sin(3x)+x(c_{3}cos(3x)+c_{4}sin(3x))

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(D^4+18D^2+81)y=0

(D^{4} + 18 D^{2} + 81) y = 0

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

Solution steps

(D^{4} + 18 D^{2} + 81) y = 0

Solve linear ODE:

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

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

y^{^{''}} + 12 y^{^{'}} + 36 y = 0, y (0) = 2, y^{^{'}} (0) = 0

2 y^{^{''}} + 8 y = 0, y^{^{'}} (0) = 1, y (0) = 0

(D^{4} - 2 D^{3} - 3 D^{2} + 4 D + 4) y = 0

40 y^{^{''}} + 7 y^{^{'}} - 3 y = 0

8 y^{^{''}} + 8 y^{^{'}} + \frac{1}{4} y = 0, y (0) = 2, y^{^{'}} (0) = 1

What is the solution for (D^4+18D^2+81)y=0 ?
The solution for (D^4+18D^2+81)y=0 is y=c_{1}cos(3x)+c_{2}sin(3x)+x(c_{3}cos(3x)+c_{4}sin(3x))