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

The solution for y^{''}-2y^'+y=e^x+e^{2x} is y=c_{1}e^x+c_{2}xe^x+(e^xx^2)/2+e^{2x}

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

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

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{x} x ^{2}}{2} + e^{2 x}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{x} x ^{2}}{2} + e^{2 x}

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{x} x ^{2}}{2} + e^{2 x}

y^{^{''}} - 2 y^{^{'}} + 5 y = e^{x} (0 = (2 x))

y^{^{''}} - 6 y^{^{'}} + 9 y = 18

y^{^{''}} - y = 8 t e^{t} + 2 e^{t}

y^{^{''}} + 18 y = 3 e^{8 x}

y^{^{''}} - 4 y^{^{'}} + 4 y = \frac{e ^{2 x}}{1 + x}

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