What is the solution for 81y^{''}+180y^'+125y=0,y(3)=6,y^'(3)=6 ?

The solution for 81y^{''}+180y^'+125y=0,y(3)=6,y^'(3)=6 is

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81y^{''}+180y^'+125y=0,y(3)=6,y^'(3)=6

81 y^{^{''}} + 180 y^{^{'}} + 125 y = 0, y (3) = 6, y^{^{'}} (3) = 6

y = e^{- \frac{10 t}{9}} - \frac{( - 30 e ^{\frac{10}{3}} ( sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + cos ( \frac{5}{3} ) ) - sin ( \frac{5}{3} ) ( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) ) cos ( \frac{5 t}{9} )}{5 cos ( \frac{5}{3} ) ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )} - \frac{( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) sin ( \frac{5 t}{9} )}{5 ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )}

Solution steps

81 y^{''} (t) + 180 y^{'} (t) + 125 y = 0

Solve linear ODE:

y = e^{- \frac{10 t}{9}} - \frac{( - 30 e ^{\frac{10}{3}} ( sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + cos ( \frac{5}{3} ) ) - sin ( \frac{5}{3} ) ( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) ) cos ( \frac{5 t}{9} )}{5 cos ( \frac{5}{3} ) ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )} - \frac{( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) sin ( \frac{5 t}{9} )}{5 ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )}

y = e^{- \frac{10 t}{9}} - \frac{( - 30 e ^{\frac{10}{3}} ( sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + cos ( \frac{5}{3} ) ) - sin ( \frac{5}{3} ) ( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) ) cos ( \frac{5 t}{9} )}{5 cos ( \frac{5}{3} ) ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )} - \frac{( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) sin ( \frac{5 t}{9} )}{5 ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )}

\frac{d}{d x} (8 cos^{2} (x))

2 x^{3} - 6 x^{2} - 2 x + 6, - x^{3} + 3 x^{2} + x - 3

y = 5 cos (7 x), y = 5 - 5 cos (7 x)

\int_{0}^{\frac{π}{16}} cos (8 x) d x

\int_{0}^{4} π (\frac{20 - 5 x}{4})^{2} d x

What is the solution for 81y^{''}+180y^'+125y=0,y(3)=6,y^'(3)=6 ?
The solution for 81y^{''}+180y^'+125y=0,y(3)=6,y^'(3)=6 is