What is the solution for y^{''}+y=3sin(2x)+xcos(2x) ?

The solution for y^{''}+y=3sin(2x)+xcos(2x) is y=c_{1}cos(x)+c_{2}sin(x)-(xcos(2x))/3-5/9 sin(2x)

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y^{''}+y=3sin(2x)+xcos(2x)

y^{^{''}} + y = 3 sin (2 x) + x cos (2 x)

y = c_{1} cos (x) + c_{2} sin (x) - \frac{x cos ( 2 x )}{3} - \frac{5}{9} sin (2 x)

Solution steps

y^{''} (x) + y = 3 sin (2 x) + x cos (2 x)

Solve linear ODE:

y = c_{1} cos (x) + c_{2} sin (x) - \frac{x cos ( 2 x )}{3} - \frac{5}{9} sin (2 x)

y = c_{1} cos (x) + c_{2} sin (x) - \frac{x cos ( 2 x )}{3} - \frac{5}{9} sin (2 x)

\frac{d ^{2} y}{d x ^{2}} - 8 \frac{d y}{d x} + 159 = 0

y^{^{''}} + 3 y^{^{'}} + 3 y = 6 sin (2 t)

y^{^{''}} - 14 y^{^{'}} + 49 y = 14 x + 3

y^{^{'''}} + y^{^{''}} - y^{^{'}} - 1 = 2 (1 + e^{- x})

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

What is the solution for y^{''}+y=3sin(2x)+xcos(2x) ?
The solution for y^{''}+y=3sin(2x)+xcos(2x) is y=c_{1}cos(x)+c_{2}sin(x)-(xcos(2x))/3-5/9 sin(2x)