What is the solution for 9y^{''}+36y^'+100y=0,y(1)=4,y^'(1)=9 ?

The solution for 9y^{''}+36y^'+100y=0,y(1)=4,y^'(1)=9 is

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9y^{''}+36y^'+100y=0,y(1)=4,y^'(1)=9

9 y^{^{''}} + 36 y^{^{'}} + 100 y = 0, y (1) = 4, y^{^{'}} (1) = 9

y = e^{- 2 t} (- \frac{( sin ( \frac{8}{3} ) ( 27 e ^{2} + 24 e ^{2} sec ( \frac{8}{3} ) cos ( \frac{8}{3} ) + 32 e ^{2} sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) ) - 8 e ^{2} ( 4 sin ^{2} ( \frac{8}{3} ) sec ( \frac{8}{3} ) + 3 cos ( \frac{8}{3} ) sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) - 3 sin ( \frac{8}{3} ) + 4 cos ( \frac{8}{3} ) ) ) cos ( \frac{8 t}{3} )}{2 cos ( \frac{8}{3} ) ( - 3 sin ( \frac{8}{3} ) + 4 cos ( \frac{8}{3} ) + 4 sin ^{2} ( \frac{8}{3} ) sec ( \frac{8}{3} ) + 3 cos ( \frac{8}{3} ) sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) )} + \frac{( 27 e ^{2} + 24 e ^{2} sec ( \frac{8}{3} ) cos ( \frac{8}{3} ) + 32 e ^{2} sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) ) sin ( \frac{8 t}{3} )}{2 ( - 3 sin ( \frac{8}{3} ) + 4 cos ( \frac{8}{3} ) + 4 sin ^{2} ( \frac{8}{3} ) sec ( \frac{8}{3} ) + 3 cos ( \frac{8}{3} ) sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) )})

Solution steps

9 y^{''} (t) + 36 y^{'} (t) + 100 y = 0

Solve linear ODE:

y = e^{- 2 t} (- \frac{( sin ( \frac{8}{3} ) ( 27 e ^{2} + 24 e ^{2} sec ( \frac{8}{3} ) cos ( \frac{8}{3} ) + 32 e ^{2} sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) ) - 8 e ^{2} ( 4 sin ^{2} ( \frac{8}{3} ) sec ( \frac{8}{3} ) + 3 cos ( \frac{8}{3} ) sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) - 3 sin ( \frac{8}{3} ) + 4 cos ( \frac{8}{3} ) ) ) cos ( \frac{8 t}{3} )}{2 cos ( \frac{8}{3} ) ( - 3 sin ( \frac{8}{3} ) + 4 cos ( \frac{8}{3} ) + 4 sin ^{2} ( \frac{8}{3} ) sec ( \frac{8}{3} ) + 3 cos ( \frac{8}{3} ) sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) )} + \frac{( 27 e ^{2} + 24 e ^{2} sec ( \frac{8}{3} ) cos ( \frac{8}{3} ) + 32 e ^{2} sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) ) sin ( \frac{8 t}{3} )}{2 ( - 3 sin ( \frac{8}{3} ) + 4 cos ( \frac{8}{3} ) + 4 sin ^{2} ( \frac{8}{3} ) sec ( \frac{8}{3} ) + 3 cos ( \frac{8}{3} ) sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) )})

y = e^{- 2 t} (- \frac{( sin ( \frac{8}{3} ) ( 27 e ^{2} + 24 e ^{2} sec ( \frac{8}{3} ) cos ( \frac{8}{3} ) + 32 e ^{2} sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) ) - 8 e ^{2} ( 4 sin ^{2} ( \frac{8}{3} ) sec ( \frac{8}{3} ) + 3 cos ( \frac{8}{3} ) sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) - 3 sin ( \frac{8}{3} ) + 4 cos ( \frac{8}{3} ) ) ) cos ( \frac{8 t}{3} )}{2 cos ( \frac{8}{3} ) ( - 3 sin ( \frac{8}{3} ) + 4 cos ( \frac{8}{3} ) + 4 sin ^{2} ( \frac{8}{3} ) sec ( \frac{8}{3} ) + 3 cos ( \frac{8}{3} ) sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) )} + \frac{( 27 e ^{2} + 24 e ^{2} sec ( \frac{8}{3} ) cos ( \frac{8}{3} ) + 32 e ^{2} sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) ) sin ( \frac{8 t}{3} )}{2 ( - 3 sin ( \frac{8}{3} ) + 4 cos ( \frac{8}{3} ) + 4 sin ^{2} ( \frac{8}{3} ) sec ( \frac{8}{3} ) + 3 cos ( \frac{8}{3} ) sec ( \frac{8}{3} ) sin ( \frac{8}{3} ) )})

y^{^{''}} = 8 y^{^{'}} - 15 y

y^{^{''}} - y^{^{'}} + a y = 0

y^{^{''''}} - 36 y^{^{'''}} + 324 y^{^{''}} = 0

x^{^{''}} + 9 x = 0, x (0) = 3, x^{^{'}} (0) = 4

y^{^{''}} - 7 y^{^{'}} + 2 y = 0

What is the solution for 9y^{''}+36y^'+100y=0,y(1)=4,y^'(1)=9 ?
The solution for 9y^{''}+36y^'+100y=0,y(1)=4,y^'(1)=9 is