What is the solution for y^{''}-2y^'-3y=4e^{-x},y(0)=4,y^'(0)=-1 ?

The solution for y^{''}-2y^'-3y=4e^{-x},y(0)=4,y^'(0)=-1 is y=e^{3x}+3e^{-x}-e^{-x}x

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y^{''}-2y^'-3y=4e^{-x},y(0)=4,y^'(0)=-1

y^{^{''}} - 2 y^{^{'}} - 3 y = 4 e^{- x}, y (0) = 4, y^{^{'}} (0) = - 1

y = e^{3 x} + 3 e^{- x} - e^{- x} x

Solution steps

y^{''} (x) - 2 y^{'} (x) - 3 y = 4 e^{- x}

Solve linear ODE:

y = e^{3 x} + 3 e^{- x} - e^{- x} x

y = e^{3 x} + 3 e^{- x} - e^{- x} x

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What is the solution for y^{''}-2y^'-3y=4e^{-x},y(0)=4,y^'(0)=-1 ?
The solution for y^{''}-2y^'-3y=4e^{-x},y(0)=4,y^'(0)=-1 is y=e^{3x}+3e^{-x}-e^{-x}x