What is the solution for (d^2y)/(dx^2)+y=sin(x) ?

The solution for (d^2y)/(dx^2)+y=sin(x) is y=c_{1}cos(x)+c_{2}sin(x)-(xcos(x))/2

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(d^2y)/(dx^2)+y=sin(x)

\frac{d ^{2} y}{d x ^{2}} + y = sin (x)

y = c_{1} cos (x) + c_{2} sin (x) - \frac{x cos ( x )}{2}

Solution steps

\frac{d ^{2} y}{d x ^{2}} + y = sin (x)

Solve linear ODE:

y = c_{1} cos (x) + c_{2} sin (x) - \frac{x cos ( x )}{2}

y = c_{1} cos (x) + c_{2} sin (x) - \frac{x cos ( x )}{2}

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

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

y^{^{''}} + y = x e^{x}

y^{^{''}} + 5 y^{^{'}} + 6 y = 24 e^{t} + 36 t^{2}

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

What is the solution for (d^2y)/(dx^2)+y=sin(x) ?
The solution for (d^2y)/(dx^2)+y=sin(x) is y=c_{1}cos(x)+c_{2}sin(x)-(xcos(x))/2