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General Solution of a Differential Equation

Solve the equ...

Question

Solve the equation:
$3 cos h 2 x + 5 cos h x = 22$

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Solution

$3 cos h 2 x + 5 cos h x = 22$

$3 (\frac{e^{2 x} + e^{- 2 x}}{2}) + 5 \frac{e^{x} + e^{- x}}{2} = 22$

$3 e^{4 x} + 5 e^{3 x} - 44 e^{2 x} + 5 e^{x} + 3 = 0$
$e^{x} = 0.982$ or $2.8464$

hence, $x$ must be $l o g (0.982)$ and $l o g (2.8464)$

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