What is the solution for y^{''}+1000y^'+250000y=0 ?

The solution for y^{''}+1000y^'+250000y=0 is y=c_{1}e^{-500t}+c_{2}te^{-500t}

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y^{''}+1000y^'+250000y=0

y^{^{''}} + 1000 y^{^{'}} + 250000 y = 0

y = c_{1} e^{- 500 t} + c_{2} t e^{- 500 t}

Solution steps

y^{''} (t) + 1000 y^{'} (t) + 250000 y = 0

Solve linear ODE:

y = c_{1} e^{- 500 t} + c_{2} t e^{- 500 t}

y = c_{1} e^{- 500 t} + c_{2} t e^{- 500 t}

y^{^{''}} + 9 y = 0, y (0) = 5, y^{^{'}} (0) = - 7

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

2 y = 3 y^{^{''}}

y^{^{''}} + 6 y^{^{'}} + 9 y = 0, y (0) = 1, y^{^{'}} (0) = 1

y^{^{''}} - k^{2} y = 0, y (0) = 0, y^{^{'}} (0) = 1

What is the solution for y^{''}+1000y^'+250000y=0 ?
The solution for y^{''}+1000y^'+250000y=0 is y=c_{1}e^{-500t}+c_{2}te^{-500t}