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

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

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12(d^2y)/(dx^2)-5(dy)/(dx)-3y=e^x

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 4 y = 16 t, y (0) = 3, y^{^{'}} (0) = - 6

y^{^{''}} - 4 y = 2

y^{^{''}} - 3 y^{^{'}} + 2 y = 9 e^{3 x}

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

y^{^{''}} + 4 y = t^{2} + 6 e^{t}

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