What is the solution for y^{''}-3y^{''}+3y^'-y=x-3e^x ?

The solution for y^{''}-3y^{''}+3y^'-y=x-3e^x is y=c_{1}e^{x/2}+c_{2}e^x-x-3+3xe^x

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y^{''}-3y^{''}+3y^'-y=x-3e^x

y^{^{''}} - 3 y^{^{''}} + 3 y^{^{'}} - y = x - 3 e^{x}

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{x} - x - 3 + 3 x e^{x}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{x} - x - 3 + 3 x e^{x}

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{x} - x - 3 + 3 x e^{x}

What is the solution for y^{''}-3y^{''}+3y^'-y=x-3e^x ?
The solution for y^{''}-3y^{''}+3y^'-y=x-3e^x is y=c_{1}e^{x/2}+c_{2}e^x-x-3+3xe^x