What is the solution for y^{'''}-y^'=2x-3x^2 ?

The solution for y^{'''}-y^'=2x-3x^2 is y=c_{1}+c_{2}e^{-x}+c_{3}e^x+x^3-x^2+6x

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y^{'''}-y^'=2x-3x^2

y^{^{'''}} - y^{^{'}} = 2 x - 3 x^{2}

y = c_{1} + c_{2} e^{- x} + c_{3} e^{x} + x^{3} - x^{2} + 6 x

Solution steps

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

Solve linear ODE:

y = c_{1} + c_{2} e^{- x} + c_{3} e^{x} + x^{3} - x^{2} + 6 x

y = c_{1} + c_{2} e^{- x} + c_{3} e^{x} + x^{3} - x^{2} + 6 x

y^{^{'''}} + y^{^{''}} - 5 y^{^{'}} + 3 y = 2 e^{(- 3 t)} + e^{t}

y^{^{''''}} + 2 y^{^{''}} + y = 11 + cos (2 t)

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

y^{^{''}} - 18 y^{^{'}} + 81 y = (x + 1) e^{x}

4 y^{^{''}} + y = 2 sec (\frac{x}{2})

What is the solution for y^{'''}-y^'=2x-3x^2 ?
The solution for y^{'''}-y^'=2x-3x^2 is y=c_{1}+c_{2}e^{-x}+c_{3}e^x+x^3-x^2+6x