What is the solution for y^{''}+4y^'+ky=0 ?

The solution for y^{''}+4y^'+ky=0 is y=c_{1}e^{(-2+sqrt(-k+4))t}+c_{2}e^{(-sqrt(-k+4)-2)t}

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y^{''}+4y^'+ky=0

y^{^{''}} + 4 y^{^{'}} + k y = 0

y = c_{1} e^{(- 2 + - k + 4) t} + c_{2} e^{(- - k + 4 - 2) t}

Solution steps

y^{''} (t) + 4 y^{'} (t) + k y = 0

Solve linear ODE:

y = c_{1} e^{(- 2 + - k + 4) t} + c_{2} e^{(- - k + 4 - 2) t}

y = c_{1} e^{(- 2 + - k + 4) t} + c_{2} e^{(- - k + 4 - 2) t}

y^{^{''}} + 4 y = 0, y (0) = 6, y^{^{'}} (0) = - 6

y^{^{''}} + 7 y^{^{'}} + 20 y = 0

2 y^{^{''}} + 14 y^{^{'}} + 12 y = 0, y (0) = 1, y^{^{'}} (0) = 0

y^{^{''}} + 4 y^{^{'}} - 8 y = 0

y^{^{''}} + 5 y^{^{'}} = 0, y (0) = 9, y^{^{'}} (0) = - 20

What is the solution for y^{''}+4y^'+ky=0 ?
The solution for y^{''}+4y^'+ky=0 is y=c_{1}e^{(-2+sqrt(-k+4))t}+c_{2}e^{(-sqrt(-k+4)-2)t}