What is the solution for 6y^{''}+y^'-y=0,y(0)=e^{x/3} ?

The solution for 6y^{''}+y^'-y=0,y(0)=e^{x/3} is y=c_{1}e^{x/3}+c_{2}e^{-x/2}

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6y^{''}+y^'-y=0,y(0)=e^{x/3}

6 y^{^{''}} + y^{^{'}} - y = 0, y (0) = e^{\frac{x}{3}}

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

Solution steps

6 y^{''} (x) + y^{'} (x) - y = 0

Solve linear ODE:

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

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

2 y^{^{''}} - 3 y^{^{'}} - 2 y = 0

y^{^{''}} - 2 y^{^{'}} + 5 y = 0, y (0) = 0, y^{^{'}} (0) = 2

y^{^{''}} - 4 y^{^{'}} + 6 y = 0

y^{^{'''}} - 9 y^{^{''}} + 28 y^{^{'}} - 20 y = 0

2 y^{^{''}} + 2 y^{^{'}} + 7 y = 0

What is the solution for 6y^{''}+y^'-y=0,y(0)=e^{x/3} ?
The solution for 6y^{''}+y^'-y=0,y(0)=e^{x/3} is y=c_{1}e^{x/3}+c_{2}e^{-x/2}