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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 4 y^{^{'}} = e^{x}

y^{^{''}} + 3 y^{^{'}} + 2 y = t H (t - 3), y (0) = 0, y^{^{'}} (0) = 1

4 \cdot y^{^{''}} - 11 \cdot y^{^{'}} - 3 y = 5 \cdot e^{3 x}

\frac{d ^{2} x}{d t ^{2}} + x = 6 cos (2 t), x (0) = 3

y^{^{''}} + y = 3 sin (2 x)

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