What is the solution for 64y^{''}+144y^'+83y=0 ?

The solution for 64y^{''}+144y^'+83y=0 is y=e^{-(9t)/8}(c_{1}cos(t/(2^{5/2)})+c_{2}sin(t/(2^{5/2)}))

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64y^{''}+144y^'+83y=0

64 y^{^{''}} + 144 y^{^{'}} + 83 y = 0

y = e^{- \frac{9 t}{8}} (c_{1} cos (\frac{t}{2 ^{\frac{5}{2}}}) + c_{2} sin (\frac{t}{2 ^{\frac{5}{2}}}))

Solution steps

64 y^{''} (t) + 144 y^{'} (t) + 83 y = 0

Solve linear ODE:

y = e^{- \frac{9 t}{8}} (c_{1} cos (\frac{t}{2 ^{\frac{5}{2}}}) + c_{2} sin (\frac{t}{2 ^{\frac{5}{2}}}))

y = e^{- \frac{9 t}{8}} (c_{1} cos (\frac{t}{2 ^{\frac{5}{2}}}) + c_{2} sin (\frac{t}{2 ^{\frac{5}{2}}}))

x^{^{''}} + 2 \cdot x^{^{'}} + x = 0

0.2 x^{^{''}} + 4 x^{^{'}} + 20 x = 0

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

(D^{4} + D^{3} - 3 D^{2} - 5 D - 2) y = 0

x^{^{''}} + 5 x^{^{'}} + 6 x = 0, x (0) = 2, x^{^{'}} (0) = 3

What is the solution for 64y^{''}+144y^'+83y=0 ?
The solution for 64y^{''}+144y^'+83y=0 is y=e^{-(9t)/8}(c_{1}cos(t/(2^{5/2)})+c_{2}sin(t/(2^{5/2)}))