The differential equation of the family of curves represented by $y = a + b x + c e^{- x}$ (where $a, b, c$ are arbitrary constants) is

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

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y^{'''} + y^{''} = 0

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y^{'''} y^{''} + y^{'} = 0

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y^{'''} + y^{''} y^{'} = 0

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Solution

The correct option is B $y^{'''} + y^{''} = 0$
$y = a + b x + c e^{- x}$
$\Rightarrow y^{'} = b - c e^{- x}$
$\Rightarrow y^{''} = c e^{- x}$
$\Rightarrow y^{'''} = - c e^{- x}$
$\Rightarrow y^{'''} + y^{''} = - c e^{- x} + c e^{- x} = 0$

Option $B$ is correct.

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