What is the solution for y^{''}+16y=8e^{4t}+8sin(2t) ?

The solution for y^{''}+16y=8e^{4t}+8sin(2t) is y=c_{1}cos(4t)+c_{2}sin(4t)+1/4 e^{4t}+2/3 sin(2t)

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y^{''}+16y=8e^{4t}+8sin(2t)

y^{^{''}} + 16 y = 8 e^{4 t} + 8 sin (2 t)

y = c_{1} cos (4 t) + c_{2} sin (4 t) + \frac{1}{4} e^{4 t} + \frac{2}{3} sin (2 t)

Solution steps

y^{''} (t) + 16 y = 8 e^{4 t} + 8 sin (2 t)

Solve linear ODE:

y = c_{1} cos (4 t) + c_{2} sin (4 t) + \frac{1}{4} e^{4 t} + \frac{2}{3} sin (2 t)

y = c_{1} cos (4 t) + c_{2} sin (4 t) + \frac{1}{4} e^{4 t} + \frac{2}{3} sin (2 t)

y^{^{''}} - 6 y^{^{'}} + 15 y = 2 sin (3 t)

y^{^{''}} - 3 y^{^{'}} + 2 y = 1, y^{^{'}} (4) = 3

y^{^{''}} + 2 y^{^{'}} - 3 y = e^{- x} (- 4 x + 3)

y^{^{''}} + 2 y^{^{'}} - 8 y = 36 e^{2 t}

y^{^{''}} + 4 y^{^{'}} + 3 y = - e^{- 6 t}, y (0) = 0, y^{^{'}} (0) = 0

What is the solution for y^{''}+16y=8e^{4t}+8sin(2t) ?
The solution for y^{''}+16y=8e^{4t}+8sin(2t) is y=c_{1}cos(4t)+c_{2}sin(4t)+1/4 e^{4t}+2/3 sin(2t)