What is the solution for x^{''}+4sqrt(2)x^'+8x=0 ?

The solution for x^{''}+4sqrt(2)x^'+8x=0 is x=c_{1}e^{-2^{1+1/2}t}+c_{2}e^{-2^{1+1/2}t}t

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x^{''}+4sqrt(2)x^'+8x=0

x^{^{''}} + 42 x^{^{'}} + 8 x = 0

x = c_{1} e^{- 2^{1 + \frac{1}{2}} t} + c_{2} e^{- 2^{1 + \frac{1}{2}} t} t

Solution steps

x^{''} (t) + 42 x^{'} (t) + 8 x = 0

Solve linear ODE:

x = c_{1} e^{- 2^{1 + \frac{1}{2}} t} + c_{2} e^{- 2^{1 + \frac{1}{2}} t} t

x = c_{1} e^{- 2^{1 + \frac{1}{2}} t} + c_{2} e^{- 2^{1 + \frac{1}{2}} t} t

y^{^{''}} + 2 y^{^{'}} + 9 y = 0, y (0) = 1, y^{^{'}} (0) = 0

y^{^{''}} - 4 y^{^{'}} + 12 y = 0, y (0) = 0, y^{^{'}} (0) = 1

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

3 x^{^{''}} - 7 x^{^{'}} + 4 x = 0

y^{^{'''}} + 6 y^{^{''}} - 27 y^{^{'}} = 0

What is the solution for x^{''}+4sqrt(2)x^'+8x=0 ?
The solution for x^{''}+4sqrt(2)x^'+8x=0 is x=c_{1}e^{-2^{1+1/2}t}+c_{2}e^{-2^{1+1/2}t}t