What is the solution for 4x^{''}+x^'+7x=0 ?

The solution for 4x^{''}+x^'+7x=0 is x=e^{-t/8}(c_{1}cos((sqrt(111)t)/8)+c_{2}sin((sqrt(111)t)/8))

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4x^{''}+x^'+7x=0

4 x^{^{''}} + x^{^{'}} + 7 x = 0

x = e^{- \frac{t}{8}} (c_{1} cos (\frac{111 t}{8}) + c_{2} sin (\frac{111 t}{8}))

Solution steps

4 x^{''} (t) + x^{'} (t) + 7 x = 0

Solve linear ODE:

x = e^{- \frac{t}{8}} (c_{1} cos (\frac{111 t}{8}) + c_{2} sin (\frac{111 t}{8}))

x = e^{- \frac{t}{8}} (c_{1} cos (\frac{111 t}{8}) + c_{2} sin (\frac{111 t}{8}))

y^{^{''}} + 0.025 \cdot y + 0.1 y^{^{'}} = 0

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

y^{^{''}} + 16 y = 0, y^{^{'}} (0) = - 2, y (0) = 2

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

8 y^{^{''}} + c y^{^{'}} + 128 y = 0, y (0) = 5, y^{^{'}} (0) = 8

What is the solution for 4x^{''}+x^'+7x=0 ?
The solution for 4x^{''}+x^'+7x=0 is x=e^{-t/8}(c_{1}cos((sqrt(111)t)/8)+c_{2}sin((sqrt(111)t)/8))