What is the solution for y^{''}-9y^'=81x^2+7,y(0)=-1,y^'(0)=2 ?

The solution for y^{''}-9y^'=81x^2+7,y(0)=-1,y^'(0)=2 is y=(-1+4/3)e^{9x}-4/3-3x^3-x^2-x

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y^{''}-9y^'=81x^2+7,y(0)=-1,y^'(0)=2

y^{^{''}} - 9 y^{^{'}} = 81 x^{2} + 7, y (0) = - 1, y^{^{'}} (0) = 2

y = (- 1 + \frac{4}{3}) e^{9 x} - \frac{4}{3} - 3 x^{3} - x^{2} - x

Solution steps

y^{''} (x) - 9 y^{'} (x) = 81 x^{2} + 7

Solve linear ODE:

y = (- 1 + \frac{4}{3}) e^{9 x} - \frac{4}{3} - 3 x^{3} - x^{2} - x

y = (- 1 + \frac{4}{3}) e^{9 x} - \frac{4}{3} - 3 x^{3} - x^{2} - x

y^{^{''}} - y^{^{'}} - 6 y = 14 sin (3)

y^{^{''}} - 4 y + 3 y = 9 x^{2}

y^{^{''}} + 6 y^{^{'}} - 40 y = - 2 e^{3 x} + cos (2 x)

y^{^{''}} + 2 y^{^{'}} + 5 y = 2 e^{(- t)} sin (2 t)

x^{^{''}} - 4 x^{^{'}} = 20 cos (5 t) - 4 sin (5 t) - 7

What is the solution for y^{''}-9y^'=81x^2+7,y(0)=-1,y^'(0)=2 ?
The solution for y^{''}-9y^'=81x^2+7,y(0)=-1,y^'(0)=2 is y=(-1+4/3)e^{9x}-4/3-3x^3-x^2-x