What is the solution for y^{''}+y^'+56y=0 ?

The solution for y^{''}+y^'+56y=0 is y=e^{-t/2}(c_{1}cos((sqrt(223)t)/2)+c_{2}sin((sqrt(223)t)/2))

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y^{''}+y^'+56y=0

y^{^{''}} + y^{^{'}} + 56 y = 0

y = e^{- \frac{t}{2}} (c_{1} cos (\frac{223 t}{2}) + c_{2} sin (\frac{223 t}{2}))

Solution steps

y^{''} (t) + y^{'} (t) + 56 y = 0

Solve linear ODE:

y = e^{- \frac{t}{2}} (c_{1} cos (\frac{223 t}{2}) + c_{2} sin (\frac{223 t}{2}))

y = e^{- \frac{t}{2}} (c_{1} cos (\frac{223 t}{2}) + c_{2} sin (\frac{223 t}{2}))

y^{^{''}} - 4 y^{^{'}} + 4 y = 0, y (0) = 5, y^{^{'}} (0) = \frac{53}{5}

y^{^{''}} - 3 y^{^{'}} + 2 y = 0, y (0) = \frac{3}{5}, y^{^{'}} (0) = - \frac{1}{5}

\frac{d ^{4} y}{d x ^{4}} + 8 \frac{d ^{2} y}{d x ^{2}} + 6 y = 0

(D^{4} - D^{2}) y = 0

\frac{1}{2} x^{^{''}} + 3 x^{^{'}} + 4 x = 0, x (0) = 3, x^{^{'}} (0) = 0

What is the solution for y^{''}+y^'+56y=0 ?
The solution for y^{''}+y^'+56y=0 is y=e^{-t/2}(c_{1}cos((sqrt(223)t)/2)+c_{2}sin((sqrt(223)t)/2))