What is the solution for y^{''}-9y^'+14y=3x^2-5sin(2x)+7xe^{6x} ?

The solution for y^{''}-9y^'+14y=3x^2-5sin(2x)+7xe^{6x} is

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y^{''}-9y^'+14y=3x^2-5sin(2x)+7xe^{6x}

y^{^{''}} - 9 y^{^{'}} + 14 y = 3 x^{2} - 5 sin (2 x) + 7 x e^{6 x}

y = c_{1} e^{7 x} + c_{2} e^{2 x} + \frac{3 x ^{2}}{14} + \frac{27 x}{98} + \frac{201}{1372} - \frac{25}{212} sin (2 x) - \frac{45}{212} cos (2 x) - \frac{7 e ^{6 x} x}{4} - \frac{21}{16} e^{6 x}

Solution steps

y^{''} (x) - 9 y^{'} (x) + 14 y = 3 x^{2} - 5 sin (2 x) + 7 x e^{6 x}

Solve linear ODE:

y = c_{1} e^{7 x} + c_{2} e^{2 x} + \frac{3 x ^{2}}{14} + \frac{27 x}{98} + \frac{201}{1372} - \frac{25}{212} sin (2 x) - \frac{45}{212} cos (2 x) - \frac{7 e ^{6 x} x}{4} - \frac{21}{16} e^{6 x}

y = c_{1} e^{7 x} + c_{2} e^{2 x} + \frac{3 x ^{2}}{14} + \frac{27 x}{98} + \frac{201}{1372} - \frac{25}{212} sin (2 x) - \frac{45}{212} cos (2 x) - \frac{7 e ^{6 x} x}{4} - \frac{21}{16} e^{6 x}

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

y^{^{''}} - 2 y^{^{'}} - 8 y = 2 cos (2 x) + (3 x + 2) e^{- 2 x}

2 y^{^{''}} - 4 y^{^{'}} + 4 y = e x sec (x)

y^{^{''}} - y^{^{'}} + \frac{1}{4} y = 6 + e^{\frac{x}{2}}

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

What is the solution for y^{''}-9y^'+14y=3x^2-5sin(2x)+7xe^{6x} ?
The solution for y^{''}-9y^'+14y=3x^2-5sin(2x)+7xe^{6x} is