What is the solution for x^{''}=-x+3x^' ?

The solution for x^{''}=-x+3x^' is x=c_{1}e^{((3+sqrt(5))t)/2}+c_{2}e^{((3-sqrt(5))t)/2}

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x^{''}=-x+3x^'

x^{^{''}} = - x + 3 x^{^{'}}

x = c_{1} e^{\frac{( 3 + 5 ) t}{2}} + c_{2} e^{\frac{( 3 - 5 ) t}{2}}

Solution steps

x^{''} (t) = - x + 3 x^{'} (t)

Solve linear ODE:

x = c_{1} e^{\frac{( 3 + 5 ) t}{2}} + c_{2} e^{\frac{( 3 - 5 ) t}{2}}

x = c_{1} e^{\frac{( 3 + 5 ) t}{2}} + c_{2} e^{\frac{( 3 - 5 ) t}{2}}

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What is the solution for x^{''}=-x+3x^' ?
The solution for x^{''}=-x+3x^' is x=c_{1}e^{((3+sqrt(5))t)/2}+c_{2}e^{((3-sqrt(5))t)/2}