What is the solution for y^{''}-2y^'=-6x^2+1 ?

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

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y^{''}-2y^'=-6x^2+1

y^{^{''}} - 2 y^{^{'}} = - 6 x^{2} + 1

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

Solution steps

y^{''} (x) - 2 y^{'} (x) = - 6 x^{2} + 1

Solve linear ODE:

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

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

y^{^{''}} + y^{^{'}} + 0.25 y = e^{x} (sin (3 x) - cos (3 x))

y^{^{''}} + 4 y^{^{'}} = t^{2} + 6 e^{t}, y (0) = 0, y^{^{'}} (0) = 9

y^{^{''}} + 4 y^{^{'}} = t^{2} + 6 e^{t}, y (0) = 0, y^{^{'}} (0) = 5

y^{^{''}} + y = t^{2} sin (t)

y^{^{'''}} - 8 y^{^{''}} + 5 y^{^{'}} + y = 5 t

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