What is the solution for y^{''}-16y=48t-24e^{-4t},y(0)=0,y^'(0)=16 ?

The solution for y^{''}-16y=48t-24e^{-4t},y(0)=0,y^'(0)=16 is y=2e^{4t}-2e^{-4t}-3t+3e^{-4t}t

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y^{''}-16y=48t-24e^{-4t},y(0)=0,y^'(0)=16

y^{^{''}} - 16 y = 48 t - 24 e^{- 4 t}, y (0) = 0, y^{^{'}} (0) = 16

y = 2 e^{4 t} - 2 e^{- 4 t} - 3 t + 3 e^{- 4 t} t

Solution steps

y^{''} (t) - 16 y = 48 t - 24 e^{- 4 t}

Solve linear ODE:

y = 2 e^{4 t} - 2 e^{- 4 t} - 3 t + 3 e^{- 4 t} t

y = 2 e^{4 t} - 2 e^{- 4 t} - 3 t + 3 e^{- 4 t} t

y^{^{''}} - 8 y^{^{'}} + 16 y = 8 x + 4

y^{^{''}} + 6 y^{^{'}} - 7 y = 3 e^{2 x} - 2 e^{3 x}

y^{^{'''}} - 4 y^{^{''}} + 13 y^{^{'}} = 104 e^{4 x}

y^{^{''}} + y^{^{'}} = 30, y (0) = - 94 t + 60 t^{2}

0.5 x^{^{''}} + 5 x^{^{'}} + 12 x = e^{- 4 t} (4 t - 2)

What is the solution for y^{''}-16y=48t-24e^{-4t},y(0)=0,y^'(0)=16 ?
The solution for y^{''}-16y=48t-24e^{-4t},y(0)=0,y^'(0)=16 is y=2e^{4t}-2e^{-4t}-3t+3e^{-4t}t