What is the solution for 10x^{''}+30x^'+20x=0 ?

The solution for 10x^{''}+30x^'+20x=0 is x=c_{1}e^{-t}+c_{2}e^{-2t}

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10x^{''}+30x^'+20x=0

10 x^{^{''}} + 30 x^{^{'}} + 20 x = 0

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

Solution steps

10 x^{''} (t) + 30 x^{'} (t) + 20 x = 0

Solve linear ODE:

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

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

y^{^{''}} + \frac{5}{2} y^{^{'}} + y = 0

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

\frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} - 4 y = 0

y^{^{''}} + (e + 1) y^{^{'}} + ey = 0

3 y^{^{''}} - y^{^{'}} = 5 y

What is the solution for 10x^{''}+30x^'+20x=0 ?
The solution for 10x^{''}+30x^'+20x=0 is x=c_{1}e^{-t}+c_{2}e^{-2t}