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

The solution for 6y^{''}-5y^'+y=3x^2+x is y=c_{1}e^{x/2}+c_{2}e^{x/3}+3x^2+31x+119

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

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

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{\frac{x}{3}} + 3 x^{2} + 31 x + 119

Solution steps

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

Solve linear ODE:

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{\frac{x}{3}} + 3 x^{2} + 31 x + 119

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{\frac{x}{3}} + 3 x^{2} + 31 x + 119

\frac{d ^{2} y}{d x ^{2}} - 5 \frac{d y}{d x} + 6 y = e^{2 x}

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

What is the solution for 6y^{''}-5y^'+y=3x^2+x ?
The solution for 6y^{''}-5y^'+y=3x^2+x is y=c_{1}e^{x/2}+c_{2}e^{x/3}+3x^2+31x+119