What is the solution for 36(d^2y)/(dx^2)+12(dy)/(dx)+y=0 ?

The solution for 36(d^2y)/(dx^2)+12(dy)/(dx)+y=0 is y=c_{1}e^{-x/6}+c_{2}e^{-x/6}x

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 6 y^{^{'}} + 8 y = 0, y (0) = - 1, y^{^{'}} (0) = 2

\frac{d ^{2} y}{d t ^{2}} - 10 \frac{d y}{d t} = 0

y^{^{''}} - 4 y + 13 y = 0

49 y^{^{'''}} + 56 y^{^{''}} + 16 y^{^{'}} = 0

\frac{d ^{3} y}{d x ^{3}} + 8 \frac{d y}{d x} = 0

What is the solution for 36(d^2y)/(dx^2)+12(dy)/(dx)+y=0 ?
The solution for 36(d^2y)/(dx^2)+12(dy)/(dx)+y=0 is y=c_{1}e^{-x/6}+c_{2}e^{-x/6}x