What is the solution for y^{''''}+8y^{''}+16y=0 ?

The solution for y^{''''}+8y^{''}+16y=0 is y=c_{1}cos(2t)+c_{2}sin(2t)+t(c_{3}cos(2t)+c_{4}sin(2t))

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y^{''''}+8y^{''}+16y=0

y^{^{''''}} + 8 y^{^{''}} + 16 y = 0

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

Solution steps

y^{(4)} (t) + 8 y^{''} (t) + 16 y = 0

Solve linear ODE:

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

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

x^{^{''}} + 6 x^{^{'}} + 9 x = 0

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

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

2 y^{^{''}} - 3 y^{^{'}} + y = 0

y^{^{''}} - 6 y = 0

What is the solution for y^{''''}+8y^{''}+16y=0 ?
The solution for y^{''''}+8y^{''}+16y=0 is y=c_{1}cos(2t)+c_{2}sin(2t)+t(c_{3}cos(2t)+c_{4}sin(2t))