What is the solution for (D^2+2)^3(D^2-3)^2y=0 ?

The solution for (D^2+2)^3(D^2-3)^2y=0 is

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(D^2+2)^3(D^2-3)^2y=0

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

y = c_{1} e^{- 3 x} + c_{2} x e^{- 3 x} + c_{3} e^{3 x} + c_{4} x e^{3 x} + c_{5} cos (2 x) + c_{6} sin (2 x) + x (c_{7} cos (2 x) + c_{8} sin (2 x)) + x^{2} (c_{9} cos (2 x) + c_{10} sin (2 x))

Solution steps

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

Solve linear ODE:

y = c_{1} e^{- 3 x} + c_{2} x e^{- 3 x} + c_{3} e^{3 x} + c_{4} x e^{3 x} + c_{5} cos (2 x) + c_{6} sin (2 x) + x (c_{7} cos (2 x) + c_{8} sin (2 x)) + x^{2} (c_{9} cos (2 x) + c_{10} sin (2 x))

y = c_{1} e^{- 3 x} + c_{2} x e^{- 3 x} + c_{3} e^{3 x} + c_{4} x e^{3 x} + c_{5} cos (2 x) + c_{6} sin (2 x) + x (c_{7} cos (2 x) + c_{8} sin (2 x)) + x^{2} (c_{9} cos (2 x) + c_{10} sin (2 x))

y^{^{'''}} + 3 y^{^{''}} - 5 y^{^{'}} - 39 y = 0, y (0) = e^{3 x}

(D^{2} + 1)^{3} y = 0

y^{^{''}} - y^{^{'}} - 12 y = 0, y (0) = 1, y^{^{'}} (0) = - 2

\frac{d ^{2} y}{d x ^{2}} - 5 \frac{d y}{d x} + 4 y = 0

y^{^{'''}} + 7 y^{^{''}} + 15 y^{^{'}} + 9 y = 0

What is the solution for (D^2+2)^3(D^2-3)^2y=0 ?
The solution for (D^2+2)^3(D^2-3)^2y=0 is