What is the solution for x^{''}-8x=0 ?

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

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x^{''}-8x=0

x^{^{''}} - 8 x = 0

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

Solution steps

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}

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

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} - 12 y = 0

y^{^{''}} - 6 y^{^{'}} + 13 y = 0, y (0) = 1, y^{^{'}} (0) = 2

(d^{2} + 4 d + 20)^{2} y = 0

y^{^{''}} + y^{^{'}} - 2 y = 0, y (0) = 6, y^{^{'}} (0) = 3

y^{^{''}} + 3 y^{^{'}} - 6 y = 0

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