What is the solution for y^{''}-12y^'+36y=18e^{6x} ?

The solution for y^{''}-12y^'+36y=18e^{6x} is y=c_{1}e^{6x}+c_{2}xe^{6x}+9x^2e^{6x}

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y^{''}-12y^'+36y=18e^{6x}

y^{^{''}} - 12 y^{^{'}} + 36 y = 18 e^{6 x}

y = c_{1} e^{6 x} + c_{2} x e^{6 x} + 9 x^{2} e^{6 x}

Solution steps

y^{''} (x) - 12 y^{'} (x) + 36 y = 18 e^{6 x}

Solve linear ODE:

y = c_{1} e^{6 x} + c_{2} x e^{6 x} + 9 x^{2} e^{6 x}

y = c_{1} e^{6 x} + c_{2} x e^{6 x} + 9 x^{2} e^{6 x}

y^{^{''}} - 5 y^{^{'}} - 6 y = e^{x}

x^{^{''}} + 3 x^{^{'}} + 2 x = 1, x (0) = 0

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

y^{^{''}} + y^{^{'}} = (3 t^{2} + 2 t) e^{t}

\frac{d ^{2} y}{d x ^{2}} + 4 y = 3 sin (2 x)

What is the solution for y^{''}-12y^'+36y=18e^{6x} ?
The solution for y^{''}-12y^'+36y=18e^{6x} is y=c_{1}e^{6x}+c_{2}xe^{6x}+9x^2e^{6x}