What is the solution for 2(d^2y)/(dx^2)-12(dy)/(dx)+18y=0 ?

The solution for 2(d^2y)/(dx^2)-12(dy)/(dx)+18y=0 is y=c_{1}e^{3x}+c_{2}xe^{3x}

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2(d^2y)/(dx^2)-12(dy)/(dx)+18y=0

2 \frac{d ^{2} y}{d x ^{2}} - 12 \frac{d y}{d x} + 18 y = 0

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

Solution steps

2 \frac{d ^{2} y}{d x ^{2}} - 12 \frac{d y}{d x} + 18 y = 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}

2 x^{^{''}} + 4 x^{^{'}} - 3 x = 0

y^{^{''''}} - 13 y^{^{''}} + 36 y = 0

y^{^{''}} + 6 y^{^{'}} + 13 y = 0, y (0) = 1, y^{^{'}} (0) = 0

9 y^{^{''}} - 16 y = 0

3 \frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} + y = 0

What is the solution for 2(d^2y)/(dx^2)-12(dy)/(dx)+18y=0 ?
The solution for 2(d^2y)/(dx^2)-12(dy)/(dx)+18y=0 is y=c_{1}e^{3x}+c_{2}xe^{3x}