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

The solution for (d^2y)/(dx^2)+y=xsin(2x) is y=c_{1}cos(x)+c_{2}sin(x)-(xsin(2x))/3-4/9 cos(2x)

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

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

y = c_{1} cos (x) + c_{2} sin (x) - \frac{x sin ( 2 x )}{3} - \frac{4}{9} cos (2 x)

Solution steps

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

Solve linear ODE:

y = c_{1} cos (x) + c_{2} sin (x) - \frac{x sin ( 2 x )}{3} - \frac{4}{9} cos (2 x)

y = c_{1} cos (x) + c_{2} sin (x) - \frac{x sin ( 2 x )}{3} - \frac{4}{9} cos (2 x)

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

y^{^{''}} + 4 y = 3 x^{2}

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

10 y^{^{''}} + 1000 y = 160 cos (8 t), y (0) = 0

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

What is the solution for (d^2y)/(dx^2)+y=xsin(2x) ?
The solution for (d^2y)/(dx^2)+y=xsin(2x) is y=c_{1}cos(x)+c_{2}sin(x)-(xsin(2x))/3-4/9 cos(2x)