What is the solution for x^{''}+3x^'+x=0 ?

The solution for x^{''}+3x^'+x=0 is x=c_{1}e^{((-3+sqrt(5))t)/2}+c_{2}e^{((-3-sqrt(5))t)/2}

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

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

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

Solution steps

x^{''} (t) + 3 x^{'} (t) + x = 0

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}}

y^{^{''}} + 16 y^{^{'}} + 100 y = 0

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

y^{^{''}} + 2 y^{^{'}} - 3 y = 0, y (0) = - 1, y^{^{'}} (0) = 3

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

25 y^{^{''}} - y = 0

What is the solution for x^{''}+3x^'+x=0 ?
The solution for x^{''}+3x^'+x=0 is x=c_{1}e^{((-3+sqrt(5))t)/2}+c_{2}e^{((-3-sqrt(5))t)/2}