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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 2 y^{^{'}} + 6 = 0

y^{^{''}} + 5 y^{^{'}} + 4 y = 20

y^{^{''}} + 4 y = cos (\frac{9}{4} t)

\frac{d ^{4} y}{d x ^{4}} + \frac{d ^{2} y}{d x ^{2}} = cos (4 x)

y^{^{''''}} - 3 y^{^{''}} - 4 y = 10 sin (x)

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