What is the solution for y^{''}-6y^'+8y=64x^2 ?

The solution for y^{''}-6y^'+8y=64x^2 is y=c_{1}e^{4x}+c_{2}e^{2x}+8x^2+12x+7

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y^{''}-6y^'+8y=64x^2

y^{^{''}} - 6 y^{^{'}} + 8 y = 64 x^{2}

y = c_{1} e^{4 x} + c_{2} e^{2 x} + 8 x^{2} + 12 x + 7

Solution steps

y^{''} (x) - 6 y^{'} (x) + 8 y = 64 x^{2}

Solve linear ODE:

y = c_{1} e^{4 x} + c_{2} e^{2 x} + 8 x^{2} + 12 x + 7

y = c_{1} e^{4 x} + c_{2} e^{2 x} + 8 x^{2} + 12 x + 7

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

y^{^{''}} - 2 y^{^{'}} = 4 e^{2 t}

4 \frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} - 3 y = cos (2 x)

y^{^{''}} + 4 y = 4, y (0) = 1, y^{^{'}} (0) = 0

y^{^{''}} + 3 y^{^{'}} + 2 y = u_{2} (t), y (0) = 6, y^{^{'}} (0) = 8

What is the solution for y^{''}-6y^'+8y=64x^2 ?
The solution for y^{''}-6y^'+8y=64x^2 is y=c_{1}e^{4x}+c_{2}e^{2x}+8x^2+12x+7