What is the solution for linear 12(d^2y)/(dx^2)-5(dy)/(dx)-3y=0 ?

The solution for linear 12(d^2y)/(dx^2)-5(dy)/(dx)-3y=0 is y=c_{1}e^{(3x)/4}+c_{2}e^{-x/3}

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linear 12(d^2y)/(dx^2)-5(dy)/(dx)-3y=0

linear

12 \frac{d ^{2} y}{d x ^{2}} - 5 \frac{d y}{d x} - 3 y = 0

y = c_{1} e^{\frac{3 x}{4}} + c_{2} e^{- \frac{x}{3}}

Solution steps

12 \frac{d ^{2} y}{d x ^{2}} - 5 \frac{d y}{d x} - 3 y = 0

Solve linear ODE:

y = c_{1} e^{\frac{3 x}{4}} + c_{2} e^{- \frac{x}{3}}

y = c_{1} e^{\frac{3 x}{4}} + c_{2} e^{- \frac{x}{3}}

\frac{d ^{2} y}{d x ^{2}} (1) - \frac{d y}{d x} (10) = 0

y^{^{'''}} - 15 y^{^{''}} + 72 y^{^{'}} - 108 y = 0

y^{^{'''}} + (- 7) y^{^{''}} + 8 y^{^{'}} + 16 y = 0

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

y^{^{''}} + a^{2} b^{2} y = 0

What is the solution for linear 12(d^2y)/(dx^2)-5(dy)/(dx)-3y=0 ?
The solution for linear 12(d^2y)/(dx^2)-5(dy)/(dx)-3y=0 is y=c_{1}e^{(3x)/4}+c_{2}e^{-x/3}