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

The solution for y^{''}+y^'-6y=5xe^{3x} is y=c_{1}e^{2x}+c_{2}e^{-3x}+(5e^{3x}x)/6-35/36 e^{3x}

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

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

y = c_{1} e^{2 x} + c_{2} e^{- 3 x} + \frac{5 e ^{3 x} x}{6} - \frac{35}{36} e^{3 x}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- 3 x} + \frac{5 e ^{3 x} x}{6} - \frac{35}{36} e^{3 x}

y = c_{1} e^{2 x} + c_{2} e^{- 3 x} + \frac{5 e ^{3 x} x}{6} - \frac{35}{36} e^{3 x}

y^{^{''}} - 6 y + 9 y = t^{2} e^{3 t}

y^{^{''}} - 12 y^{^{'}} + 36 y = 14 e^{6 x}

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

y^{^{'''}} - y^{^{''}} - 25 y^{^{'}} + 25 y = 4 - e^{x} + e^{5 x}

y^{^{''}} + 4 y = 3, y (0) = 0, y^{^{'}} (0) = 0

What is the solution for y^{''}+y^'-6y=5xe^{3x} ?
The solution for y^{''}+y^'-6y=5xe^{3x} is y=c_{1}e^{2x}+c_{2}e^{-3x}+(5e^{3x}x)/6-35/36 e^{3x}