What is the solution for y^{''}+2y+5y=5sin(2t) ?

The solution for y^{''}+2y+5y=5sin(2t) is y=c_{1}cos(sqrt(7)t)+c_{2}sin(sqrt(7)t)+5/3 sin(2t)

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y^{''}+2y+5y=5sin(2t)

y^{^{''}} + 2 y + 5 y = 5 sin (2 t)

y = c_{1} cos (7 t) + c_{2} sin (7 t) + \frac{5}{3} sin (2 t)

Solution steps

y^{''} (t) + 2 y + 5 y = 5 sin (2 t)

Solve linear ODE:

y = c_{1} cos (7 t) + c_{2} sin (7 t) + \frac{5}{3} sin (2 t)

y = c_{1} cos (7 t) + c_{2} sin (7 t) + \frac{5}{3} sin (2 t)

y^{^{''}} + y = - 8 e^{3 t}

y^{^{''}} - 3 y^{^{'}} - 4 y = - 8 e^{t} (cos (2 t))

y^{^{''}} + y^{^{'}} - 12 y = - 32 t + 24 t^{2}

\frac{d ^{2} y}{d x ^{2}} + y = 3 + e^{- x} + 5 e^{2 x}

y^{^{''}} + 20 y^{^{'}} + 200 y = 320

What is the solution for y^{''}+2y+5y=5sin(2t) ?
The solution for y^{''}+2y+5y=5sin(2t) is y=c_{1}cos(sqrt(7)t)+c_{2}sin(sqrt(7)t)+5/3 sin(2t)