What is the solution for y^{''}+6y^'+10y=5t-2 ?

The solution for y^{''}+6y^'+10y=5t-2 is y=e^{-3t}(c_{1}cos(t)+c_{2}sin(t))+t/2-1/2

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y^{''}+6y^'+10y=5t-2

y^{^{''}} + 6 y^{^{'}} + 10 y = 5 t - 2

y = e^{- 3 t} (c_{1} cos (t) + c_{2} sin (t)) + \frac{t}{2} - \frac{1}{2}

Solution steps

y^{''} (t) + 6 y^{'} (t) + 10 y = 5 t - 2

Solve linear ODE:

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y = e^{- 3 t} (c_{1} cos (t) + c_{2} sin (t)) + \frac{t}{2} - \frac{1}{2}

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y^{2} e^{t} + (e^{t} + 2) y^{^{'}} = 0

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What is the solution for y^{''}+6y^'+10y=5t-2 ?
The solution for y^{''}+6y^'+10y=5t-2 is y=e^{-3t}(c_{1}cos(t)+c_{2}sin(t))+t/2-1/2