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Definition of Function

If y=e4x+2e-x...

Question

If $y = e^{4 x} + 2 e^{- x}$ , then the value of $y^{'''} - 13 y^{'} - 12 y + 7$ is equal to

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Solution

Given, $y = e^{4 x} + 2 e^{- x}$
Differentiating w.r.t. $x$
$y^{'} = 4 e^{4 x} - 2 e^{- x}$
Differentiating again w.r.t. $x,$
$y^{''} = 16 e^{4 x} + 2 e^{- x}$
Differentiating again w.r.t. $x,$
$y^{'''} = 64 e^{4 x} - 2 e^{- x}$
Now, $y^{'''} - 13 y^{'} - 12 y + 7 = 64 e^{4 x} - 2 e^{- x} - 13 (4 e^{4 x} - 2 e^{- x}) - 12 (e^{4 x} + 2 e^{- x}) + 7$
$= 64 e^{4 x} - 2 e^{- x} - 52 e^{4 x} + 26 e^{- x} - 12 e^{4 x} - 24 e^{- x} + 7$
$\Rightarrow y^{'''} - 13 y^{'} - 12 y + 7 = 7$

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