What is the solution for y^{''}-12y^'+36y=8e^{6x} ?

The solution for y^{''}-12y^'+36y=8e^{6x} is y=c_{1}e^{6x}+c_{2}xe^{6x}+4x^2e^{6x}

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y^{''}-12y^'+36y=8e^{6x}

y^{^{''}} - 12 y^{^{'}} + 36 y = 8 e^{6 x}

y = c_{1} e^{6 x} + c_{2} x e^{6 x} + 4 x^{2} e^{6 x}

Solution steps

y^{''} (x) - 12 y^{'} (x) + 36 y = 8 e^{6 x}

Solve linear ODE:

y = c_{1} e^{6 x} + c_{2} x e^{6 x} + 4 x^{2} e^{6 x}

y = c_{1} e^{6 x} + c_{2} x e^{6 x} + 4 x^{2} e^{6 x}

x^{^{''}} - 5 x^{^{'}} + 6 x = 12, x (0) = 2, x^{^{'}} (0) = 0

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

y^{^{''}} - 2 y^{^{'}} + y = e^{t - 1}, y (1) = 0, y^{^{'}} (1) = 5

y^{^{''}} - y^{^{'}} = x e^{2 x}

y^{^{''}} + 4 y = - 8 sin (2 t)

What is the solution for y^{''}-12y^'+36y=8e^{6x} ?
The solution for y^{''}-12y^'+36y=8e^{6x} is y=c_{1}e^{6x}+c_{2}xe^{6x}+4x^2e^{6x}