What is the solution for y^{''}-4y^'+85y=0 ?

The solution for y^{''}-4y^'+85y=0 is y=e^{2t}(c_{1}cos(9t)+c_{2}sin(9t))

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y^{''}-4y^'+85y=0

y^{^{''}} - 4 y^{^{'}} + 85 y = 0

y = e^{2 t} (c_{1} cos (9 t) + c_{2} sin (9 t))

Solution steps

y^{''} (t) - 4 y^{'} (t) + 85 y = 0

Solve linear ODE:

y = e^{2 t} (c_{1} cos (9 t) + c_{2} sin (9 t))

y = e^{2 t} (c_{1} cos (9 t) + c_{2} sin (9 t))

y^{^{'''}} - 18 y^{^{''}} + 89 y^{^{'}} - 132 y = 0

y^{^{''}} + 0.75 y^{^{'}} + 0.3 y = 0, y^{^{'}} (0) = 0

16 y^{^{''}} - 24 y^{^{'}} + 9 y = 0, y^{1} (0) = e^{\frac{3 x}{4}}

400 y^{^{''}} - 9 y = 0

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

What is the solution for y^{''}-4y^'+85y=0 ?
The solution for y^{''}-4y^'+85y=0 is y=e^{2t}(c_{1}cos(9t)+c_{2}sin(9t))