What is the solution for y^{''}+y=2x*sin(x) ?

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

2 y^{^{''}} - 4 y^{^{'}} + 3 y = 2 e^{3 x} - 6

y^{^{''}} - 5 y = (x - 1) sin (x) + (x + 1) cos (x)

y^{^{''}} + 9 y^{^{'}} + 8 y = 3 e^{5 x}

y^{^{''}} - 6 y^{^{'}} + 25 y = 2 sin (t) - cos (t)

y^{^{''}} - 2 y^{^{'}} - 3 y = e^{3 t}

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