What is the solution for y^{''}+16y=e^{4t}+9cos(2t) ?

The solution for y^{''}+16y=e^{4t}+9cos(2t) is y=c_{1}cos(4t)+c_{2}sin(4t)+1/32 e^{4t}+3/4 cos(2t)

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y^{''}+16y=e^{4t}+9cos(2t)

y^{^{''}} + 16 y = e^{4 t} + 9 cos (2 t)

y = c_{1} cos (4 t) + c_{2} sin (4 t) + \frac{1}{32} e^{4 t} + \frac{3}{4} cos (2 t)

Solution steps

y^{''} (t) + 16 y = e^{4 t} + 9 cos (2 t)

Solve linear ODE:

y = c_{1} cos (4 t) + c_{2} sin (4 t) + \frac{1}{32} e^{4 t} + \frac{3}{4} cos (2 t)

y = c_{1} cos (4 t) + c_{2} sin (4 t) + \frac{1}{32} e^{4 t} + \frac{3}{4} cos (2 t)

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

\frac{d ^{2} y}{d x ^{2}} - 4 y^{^{'}} - 12 y = 2 x + 6

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

x^{^{''}} + 5 x^{^{'}} + 4 x = 8 e^{t}

y^{^{''}} + 20 y^{^{'}} + 200 y = 1

What is the solution for y^{''}+16y=e^{4t}+9cos(2t) ?
The solution for y^{''}+16y=e^{4t}+9cos(2t) is y=c_{1}cos(4t)+c_{2}sin(4t)+1/32 e^{4t}+3/4 cos(2t)