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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - y = 2 ln (x)

y^{^{''}} - 2 y^{^{'}} + y = 25 cos (2 x)

\frac{d ^{2} y}{d x ^{2}} = a y + b

y^{^{''}} + 6 y^{^{'}} + 5 y = 10 cos (5 t)

y^{^{''}} - 4 y^{^{'}} - 12 y = sin (3 t)

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