What is the solution for (d^2y)/(dx^2)+7(dy)/(dx)+12y=0 ?

The solution for (d^2y)/(dx^2)+7(dy)/(dx)+12y=0 is y=c_{1}e^{-3x}+c_{2}e^{-4x}

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(d^2y)/(dx^2)+7(dy)/(dx)+12y=0

\frac{d ^{2} y}{d x ^{2}} + 7 \frac{d y}{d x} + 12 y = 0

y = c_{1} e^{- 3 x} + c_{2} e^{- 4 x}

Solution steps

\frac{d ^{2} y}{d x ^{2}} + 7 \frac{d y}{d x} + 12 y = 0

Solve linear ODE:

y = c_{1} e^{- 3 x} + c_{2} e^{- 4 x}

y = c_{1} e^{- 3 x} + c_{2} e^{- 4 x}

y^{^{''}} + 12 y^{^{'}} + 37 y = 0

y^{^{'''}} - 5 y^{^{''}} - 25 y^{^{'}} + 125 y = 0

y^{^{''}} - y^{^{'}} - 12 y = 0, y (1) = 0, y^{^{'}} (1) = 1

25 y^{^{''}} + 9 y = 0

(D^{2} - 2 D + 2) y = 0

What is the solution for (d^2y)/(dx^2)+7(dy)/(dx)+12y=0 ?
The solution for (d^2y)/(dx^2)+7(dy)/(dx)+12y=0 is y=c_{1}e^{-3x}+c_{2}e^{-4x}