What is the solution for 81y^{''}+72y^'+16y=0 ?

The solution for 81y^{''}+72y^'+16y=0 is y=c_{1}e^{-(4t)/9}+c_{2}e^{-(4t)/9}t

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81y^{''}+72y^'+16y=0

81 y^{^{''}} + 72 y^{^{'}} + 16 y = 0

y = c_{1} e^{- \frac{4 t}{9}} + c_{2} e^{- \frac{4 t}{9}} t

Solution steps

81 y^{''} (t) + 72 y^{'} (t) + 16 y = 0

Solve linear ODE:

y = c_{1} e^{- \frac{4 t}{9}} + c_{2} e^{- \frac{4 t}{9}} t

y = c_{1} e^{- \frac{4 t}{9}} + c_{2} e^{- \frac{4 t}{9}} t

y^{^{''}} + y^{^{'}} + 1.25 y = 0

121 y^{^{''}} + 88 y^{^{'}} + 16 y = 0

y^{^{'''}} - 4 y^{^{''}} + 3 y^{^{'}} = 0

y^{^{''}} - y^{^{'}} - 56 y = 0, y (0) = - 5, y^{^{'}} (0) = 5

y^{^{''}} + 36 y = 0, y (0) = 0, y^{^{'}} (0) = \frac{24}{5}

What is the solution for 81y^{''}+72y^'+16y=0 ?
The solution for 81y^{''}+72y^'+16y=0 is y=c_{1}e^{-(4t)/9}+c_{2}e^{-(4t)/9}t