What is the solution for (d^2y)/(dx^2)+8(dy)/(dx)+25y=0 ?

The solution for (d^2y)/(dx^2)+8(dy)/(dx)+25y=0 is y=e^{-4x}(c_{1}cos(3x)+c_{2}sin(3x))

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(d^2y)/(dx^2)+8(dy)/(dx)+25y=0

\frac{d ^{2} y}{d x ^{2}} + 8 \frac{d y}{d x} + 25 y = 0

y = e^{- 4 x} (c_{1} cos (3 x) + c_{2} sin (3 x))

Solution steps

\frac{d ^{2} y}{d x ^{2}} + 8 \frac{d y}{d x} + 25 y = 0

Solve linear ODE:

y = e^{- 4 x} (c_{1} cos (3 x) + c_{2} sin (3 x))

y = e^{- 4 x} (c_{1} cos (3 x) + c_{2} sin (3 x))

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

3 y^{^{''}} + 7 y^{^{'}} + 2 y = 0

y^{^{''}} + 9 y = 0, y (0) = 0, y^{^{'}} (0) = 3

y^{^{''}} + 9 y = 0, y (0) = 0, y^{^{'}} (0) = 1

(d^{4} + 1) y = 0

What is the solution for (d^2y)/(dx^2)+8(dy)/(dx)+25y=0 ?
The solution for (d^2y)/(dx^2)+8(dy)/(dx)+25y=0 is y=e^{-4x}(c_{1}cos(3x)+c_{2}sin(3x))