What is the solution for (D^2-2D+1)y=e^{2x}(e^x+1)^{-2} ?

The solution for (D^2-2D+1)y=e^{2x}(e^x+1)^{-2} is y=c_{1}e^x+c_{2}xe^x+e^x(-x+ln(e^x+1))

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 6 y^{^{'}} + 9 y = t^{2} e^{3} t

y^{^{''}} - 10 y^{^{'}} + 25 y = 132 x^{2} (e^{(5 x)})

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

y^{^{''}} - 3 y^{^{'}} = 2 e^{2 x} sin (x)

y^{^{''}} + y^{^{'}} + y = e^{- t} + sin (2 t)

What is the solution for (D^2-2D+1)y=e^{2x}(e^x+1)^{-2} ?
The solution for (D^2-2D+1)y=e^{2x}(e^x+1)^{-2} is y=c_{1}e^x+c_{2}xe^x+e^x(-x+ln(e^x+1))