What is the solution for y^{''}-5y+6y=e^{3x}-x^2 ?

The solution for y^{''}-5y+6y=e^{3x}-x^2 is y=c_{1}cos(x)+c_{2}sin(x)+1/10 e^{3x}-x^2+2

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y^{''}-5y+6y=e^{3x}-x^2

y^{^{''}} - 5 y + 6 y = e^{3 x} - x^{2}

y = c_{1} cos (x) + c_{2} sin (x) + \frac{1}{10} e^{3 x} - x^{2} + 2

Solution steps

y^{''} (x) - 5 y + 6 y = e^{3 x} - x^{2}

Solve linear ODE:

y = c_{1} cos (x) + c_{2} sin (x) + \frac{1}{10} e^{3 x} - x^{2} + 2

y = c_{1} cos (x) + c_{2} sin (x) + \frac{1}{10} e^{3 x} - x^{2} + 2

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

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

y^{^{'''}} + y^{^{''}} + y^{^{'}} + y = e^{- t} + 11 t

y^{^{'''}} + y^{^{''}} + y^{^{'}} + y = e^{- t} + 12 t

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

What is the solution for y^{''}-5y+6y=e^{3x}-x^2 ?
The solution for y^{''}-5y+6y=e^{3x}-x^2 is y=c_{1}cos(x)+c_{2}sin(x)+1/10 e^{3x}-x^2+2