What is the solution for y^{''}+y=8cos(x) ?

The solution for y^{''}+y=8cos(x) is y=c_{1}cos(x)+c_{2}sin(x)+4xsin(x)

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y^{''}+y=8cos(x)

y^{^{''}} + y = 8 cos (x)

y = c_{1} cos (x) + c_{2} sin (x) + 4 x sin (x)

Solution steps

y^{''} (x) + y = 8 cos (x)

Solve linear ODE:

y = c_{1} cos (x) + c_{2} sin (x) + 4 x sin (x)

y = c_{1} cos (x) + c_{2} sin (x) + 4 x sin (x)

y^{^{''}} - y^{^{'}} - 2 y = 6 x + 6

y^{^{''}} + y^{^{'}} = 8

x^{^{''}} - 4 x^{^{'}} + 4 x = e^{t} sin (t)

x^{^{''}} - 8 x^{^{'}} + 16 x = 3 t e^{4 t}

\frac{d ^{2} y}{d x ^{2}} + 3 y^{^{'}} + 2 = 0

What is the solution for y^{''}+y=8cos(x) ?
The solution for y^{''}+y=8cos(x) is y=c_{1}cos(x)+c_{2}sin(x)+4xsin(x)