What is the solution for 6(d^2y)/(dx^2)-(dy)/(dx)-y=0 ?

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

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6(d^2y)/(dx^2)-(dy)/(dx)-y=0

6 \frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} - y = 0

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

Solution steps

6 \frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} - y = 0

Solve linear ODE:

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

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

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

(D^{2} + 1) y = 0

2 \frac{d ^{2} y}{d x ^{2}} - 12 \frac{d y}{d x} + 18 y = 0

2 x^{^{''}} + 4 x^{^{'}} - 3 x = 0

y^{^{''''}} - 13 y^{^{''}} + 36 y = 0

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