What is the solution for y^{''}-y^'-6y=3x-1 ?

The solution for y^{''}-y^'-6y=3x-1 is y=c_{1}e^{3x}+c_{2}e^{-2x}-x/2+1/4

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y^{''}-y^'-6y=3x-1

y^{^{''}} - y^{^{'}} - 6 y = 3 x - 1

y = c_{1} e^{3 x} + c_{2} e^{- 2 x} - \frac{x}{2} + \frac{1}{4}

Solution steps

y^{''} (x) - y^{'} (x) - 6 y = 3 x - 1

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} e^{- 2 x} - \frac{x}{2} + \frac{1}{4}

y = c_{1} e^{3 x} + c_{2} e^{- 2 x} - \frac{x}{2} + \frac{1}{4}

(d^{2} + 3 d + 2) \cdot y = e^{e^{x}}

(d^{2} + 9) y = sin^{2} (x)

(d^{2} - 1) y = x^{2} + cos^{2} (x)

(d^{2} + d - 6) y = e^{2} x \cdot sin^{3} (x)

y^{^{''}} + y^{^{'}} - 6 y = 6 x^{2} - 2 x - 2

What is the solution for y^{''}-y^'-6y=3x-1 ?
The solution for y^{''}-y^'-6y=3x-1 is y=c_{1}e^{3x}+c_{2}e^{-2x}-x/2+1/4