What is the solution for y^{''}-2y^'-3y=12e^x-3 ?

The solution for y^{''}-2y^'-3y=12e^x-3 is y=c_{1}e^{3x}+c_{2}e^{-x}-3e^x+1

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y^{''}-2y^'-3y=12e^x-3

y^{^{''}} - 2 y^{^{'}} - 3 y = 12 e^{x} - 3

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

Solution steps

y^{''} (x) - 2 y^{'} (x) - 3 y = 12 e^{x} - 3

Solve linear ODE:

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

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

y^{^{''}} + y^{^{'}} + y = x sin (x)

y^{^{''}} - 2 y^{^{'}} - 3 y = 8 e^{x} - 9

5 y^{^{''}} - 10 y^{^{'}} + 10 y = e^{x} sec (x)

y^{^{''}} - 16 y = 32 + 45 e^{x}

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

What is the solution for y^{''}-2y^'-3y=12e^x-3 ?
The solution for y^{''}-2y^'-3y=12e^x-3 is y=c_{1}e^{3x}+c_{2}e^{-x}-3e^x+1