What is the solution for 36y^{''}+108y^'+97y=0,y^'(1)=3,y(1)=4 ?

The solution for 36y^{''}+108y^'+97y=0,y^'(1)=3,y(1)=4 is

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36y^{''}+108y^'+97y=0,y^'(1)=3,y(1)=4

36 y^{^{''}} + 108 y^{^{'}} + 97 y = 0, y^{^{'}} (1) = 3, y (1) = 4

y = e^{- \frac{3 t}{2}} \frac{( 36 e ^{\frac{3}{2}} + \frac{9 e ^{\frac{3}{2}} s i n ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} - \frac{4 e ^{\frac{3}{2}} c o s ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} ) cos ( \frac{2 t}{3} )}{2 ( - 9 cos ( \frac{2}{3} ) - 4 sin ( \frac{2}{3} ) )} + \frac{e ^{\frac{3}{2}} ( 27 cos ( \frac{2}{3} ) + 8 sin ( \frac{2}{3} ) ) sin ( \frac{2 t}{3} )}{2 ( cos ^{2} ( \frac{2}{3} ) + sin ^{2} ( \frac{2}{3} ) )}

Solution steps

36 y^{''} (t) + 108 y^{'} (t) + 97 y = 0

Solve linear ODE:

y = e^{- \frac{3 t}{2}} \frac{( 36 e ^{\frac{3}{2}} + \frac{9 e ^{\frac{3}{2}} s i n ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} - \frac{4 e ^{\frac{3}{2}} c o s ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} ) cos ( \frac{2 t}{3} )}{2 ( - 9 cos ( \frac{2}{3} ) - 4 sin ( \frac{2}{3} ) )} + \frac{e ^{\frac{3}{2}} ( 27 cos ( \frac{2}{3} ) + 8 sin ( \frac{2}{3} ) ) sin ( \frac{2 t}{3} )}{2 ( cos ^{2} ( \frac{2}{3} ) + sin ^{2} ( \frac{2}{3} ) )}

y = e^{- \frac{3 t}{2}} \frac{( 36 e ^{\frac{3}{2}} + \frac{9 e ^{\frac{3}{2}} s i n ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} - \frac{4 e ^{\frac{3}{2}} c o s ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} ) cos ( \frac{2 t}{3} )}{2 ( - 9 cos ( \frac{2}{3} ) - 4 sin ( \frac{2}{3} ) )} + \frac{e ^{\frac{3}{2}} ( 27 cos ( \frac{2}{3} ) + 8 sin ( \frac{2}{3} ) ) sin ( \frac{2 t}{3} )}{2 ( cos ^{2} ( \frac{2}{3} ) + sin ^{2} ( \frac{2}{3} ) )}

25 y^{^{''}} - 80 y^{^{'}} + 64 y = 0

5 y^{^{''}} + 4 y^{^{'}} + y = 0, y (0) = 1, y^{^{'}} (0) = 0

x^{^{''}} - x^{^{'}} - 6 x = 0, x (0) = 2, x^{^{'}} (0) = - 1

x^{^{''}} + 48 x = 0

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

What is the solution for 36y^{''}+108y^'+97y=0,y^'(1)=3,y(1)=4 ?
The solution for 36y^{''}+108y^'+97y=0,y^'(1)=3,y(1)=4 is