What is the solution for (d^2y)/(dx^2)+10((dy)/(dx))+25y=0 ?

The solution for (d^2y)/(dx^2)+10((dy)/(dx))+25y=0 is y=c_{1}e^{-5x}+c_{2}xe^{-5x}

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(d^2y)/(dx^2)+10((dy)/(dx))+25y=0

\frac{d ^{2} y}{d x ^{2}} + 10 (\frac{d y}{d x}) + 25 y = 0

y = c_{1} e^{- 5 x} + c_{2} x e^{- 5 x}

Solution steps

\frac{d ^{2} y}{d x ^{2}} + 10 (\frac{d y}{d x}) + 25 y = 0

Solve linear ODE:

y = c_{1} e^{- 5 x} + c_{2} x e^{- 5 x}

y = c_{1} e^{- 5 x} + c_{2} x e^{- 5 x}

x^{^{''}} + 8 x^{^{'}} + 80 x = 0

y^{^{''}} + 4 y^{^{'}} + 4 y = 0, y (- 1) = 5, y^{^{'}} (- 1) = 4

r^{^{''}} - 4 r^{^{'}} - 53 r = 0

(D^{3} + 2 D^{2} + D + 2) y = 0

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

What is the solution for (d^2y)/(dx^2)+10((dy)/(dx))+25y=0 ?
The solution for (d^2y)/(dx^2)+10((dy)/(dx))+25y=0 is y=c_{1}e^{-5x}+c_{2}xe^{-5x}