What is the solution for (d^2y)/(dx^2)+4(dy)/(dx)+8y=0 ?

The solution for (d^2y)/(dx^2)+4(dy)/(dx)+8y=0 is y=e^{-2x}(c_{1}cos(2x)+c_{2}sin(2x))

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(d^2y)/(dx^2)+4(dy)/(dx)+8y=0

\frac{d ^{2} y}{d x ^{2}} + 4 \frac{d y}{d x} + 8 y = 0

y = e^{- 2 x} (c_{1} cos (2 x) + c_{2} sin (2 x))

Solution steps

\frac{d ^{2} y}{d x ^{2}} + 4 \frac{d y}{d x} + 8 y = 0

Solve linear ODE:

y = e^{- 2 x} (c_{1} cos (2 x) + c_{2} sin (2 x))

y = e^{- 2 x} (c_{1} cos (2 x) + c_{2} sin (2 x))

y^{^{''}} - k \cdot y = 0

(6 d^{3} + 5 d^{2}) y = 0

y^{^{'''}} + 7 y^{^{''}} + 14 y^{^{'}} + 8 y = 0

\frac{d ^{4} y}{d t ^{4}} + 2 y^{^{''}} + y = 0

y^{^{''}} - 5 y^{^{'}} + 6 y = 0, y (0) = 1, y^{^{'}} (0) = - 5

What is the solution for (d^2y)/(dx^2)+4(dy)/(dx)+8y=0 ?
The solution for (d^2y)/(dx^2)+4(dy)/(dx)+8y=0 is y=e^{-2x}(c_{1}cos(2x)+c_{2}sin(2x))