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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{'''}} = - y

y^{^{''}} + 0 y + 64 y = 0, y (0) = 1, y^{^{'}} (0) = 0

\frac{d ^{2} y}{d x ^{2}} - 8 \frac{d y}{d x} = 0

y^{^{''}} + 8 y^{^{'}} + 25 y = 0, y^{^{'}} (0) = 2

\frac{d ^{2} y}{d x ^{2}} + 6 \frac{d y}{d x} + 5 y = 0

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