What is the solution for x^{''}-x^'-3/4 x=0 ?

The solution for x^{''}-x^'-3/4 x=0 is x=c_{1}e^{(3t)/2}+c_{2}e^{-t/2}

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x^{''}-x^'-3/4 x=0

x^{^{''}} - x^{^{'}} - \frac{3}{4} x = 0

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

Solution steps

x^{''} (t) - x^{'} (t) - \frac{3}{4} x = 0

Solve linear ODE:

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

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

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

y^{^{''}} + 8 y^{^{'}} + 17 y = 0, y (0) = 1, y^{^{'}} (0) = - 5

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

y^{^{''}} - 10 y^{^{'}} + 34 y = 0, y (\frac{π}{6}) = 1, y (0) = 4

10 y^{^{''}} + 30 y^{^{'}} + 20 y = 0, y (0) = - 7, y^{^{'}} (0) = - 2

What is the solution for x^{''}-x^'-3/4 x=0 ?
The solution for x^{''}-x^'-3/4 x=0 is x=c_{1}e^{(3t)/2}+c_{2}e^{-t/2}