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

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

3 \frac{d ^{2} y}{d x ^{2}} + 4 \frac{d y}{d x} - 2 y = 0

9 y^{^{''}} - 18 y^{^{'}} + 9 y = 0, y (0) = 2, y^{^{'}} (0) = 3

x^{^{''}} - 3 x^{^{'}} + 2 x = 0, x (0) = 4, x^{^{'}} (0) = 6

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))