What is the solution for (d^2y)/(dx^2)+12*(dy)/(dx)+27y=0 ?

The solution for (d^2y)/(dx^2)+12*(dy)/(dx)+27y=0 is y=c_{1}e^{-3x}+c_{2}e^{-9x}

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(d^2y)/(dx^2)+12*(dy)/(dx)+27y=0

\frac{d ^{2} y}{d x ^{2}} + 12 \cdot \frac{d y}{d x} + 27 y = 0

y = c_{1} e^{- 3 x} + c_{2} e^{- 9 x}

Solution steps

\frac{d ^{2} y}{d x ^{2}} + 12 \cdot \frac{d y}{d x} + 27 y = 0

Solve linear ODE:

y = c_{1} e^{- 3 x} + c_{2} e^{- 9 x}

y = c_{1} e^{- 3 x} + c_{2} e^{- 9 x}

y^{^{'''}} - 7 y^{^{''}} + 10 y^{^{'}} = 0

2 y^{^{''}} - 5 y^{^{'}} - 3 y = 0

y^{^{'''}} + 6 y^{^{''}} - 16 y^{^{'}} = 0

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

625 x^{^{''}} + 100 x^{^{'}} + 3 x = 0, x (0) = 25, x^{^{'}} (0) = - 2

What is the solution for (d^2y)/(dx^2)+12*(dy)/(dx)+27y=0 ?
The solution for (d^2y)/(dx^2)+12*(dy)/(dx)+27y=0 is y=c_{1}e^{-3x}+c_{2}e^{-9x}