What is the solution for y^{''}-2y^'+y=(e^x-1)^2,y(0)=0,y^'(0)=1 ?

The solution for y^{''}-2y^'+y=(e^x-1)^2,y(0)=0,y^'(0)=1 is y=-2e^x+e^xx+e^{2x}-e^xx^2+1

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y^{''}-2y^'+y=(e^x-1)^2,y(0)=0,y^'(0)=1

y^{^{''}} - 2 y^{^{'}} + y = (e^{x} - 1)^{2}, y (0) = 0, y^{^{'}} (0) = 1

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

Solution steps

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

Solve linear ODE:

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

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

(D^{2} + 3 D - 4) y = 15 e^{x}

5 y^{^{''}} + y^{^{'}} = - 10 x, y (0) = 0, y^{^{'}} (0) = - 20

2 y^{^{''}} + 3 y^{^{'}} + y = g t

y^{^{''}} - y = 2 t, y (0) = - 2 t

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

What is the solution for y^{''}-2y^'+y=(e^x-1)^2,y(0)=0,y^'(0)=1 ?
The solution for y^{''}-2y^'+y=(e^x-1)^2,y(0)=0,y^'(0)=1 is y=-2e^x+e^xx+e^{2x}-e^xx^2+1