What is the solution for ((d^2y)/(dx^2)-6(dy)/(dx)+9y)^2=0 ?

The solution for ((d^2y)/(dx^2)-6(dy)/(dx)+9y)^2=0 is y=c_{1}e^{3x}+c_{2}xe^{3x}

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((d^2y)/(dx^2)-6(dy)/(dx)+9y)^2=0

(\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 9 y)^{2} = 0

y = c_{1} e^{3 x} + c_{2} x e^{3 x}

Solution steps

(\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 9 y)^{2} = 0

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} x e^{3 x}

y = c_{1} e^{3 x} + c_{2} x e^{3 x}

y^{^{''}} + 12 y^{^{'}} + 36 y = 0, y^{^{'}} (0) = 0, y (0) = 5

100 y^{^{''}} + 125 y^{^{'}} + 100 y = 0

5 y^{^{''}} + 25 y^{^{'}} + 100 y = 0

y^{^{''}} + 49 y = 0, y (0) = 6, y^{^{'}} (0) = - 6

3 y^{^{'''}} + 11 y^{^{''}} + 24 y^{^{'}} + 12 y = 0

What is the solution for ((d^2y)/(dx^2)-6(dy)/(dx)+9y)^2=0 ?
The solution for ((d^2y)/(dx^2)-6(dy)/(dx)+9y)^2=0 is y=c_{1}e^{3x}+c_{2}xe^{3x}