find-all-functions-f-2-derivable-verify-f-x-2-f-x-2-1-and-f-0-1-

Question Number 90970 by abdomathmax last updated on 27/Apr/20

f i n d a l l f u n c t i o n s f (2 \times d e r i v a b l e) v e r i f y

{(f^{^{'}} (x))}^{2} - {(f (x))}^{2} = 1 a n d f^{^{'}} (0) = 1

Commented by mr W last updated on 27/Apr/20

y^{'} = \frac{d y}{d x} = \pm \sqrt{1 + y^{2}}

\frac{d y}{\sqrt{1 + y^{2}}} = \pm d x

\int \frac{d y}{\sqrt{1 + y^{2}}} = \pm \int d x

\ln (y + \sqrt{1 + y^{2}}) = \sinh^{- 1} y = \pm (x + C)

\Rightarrow y = \pm \sinh (x + C)

y^{'} = \pm \cosh (x + C)

\pm \cosh (0 + C) = 1 \Rightarrow C = 0

\Rightarrow y = \pm \sinh (x)

Commented by mathmax by abdo last updated on 28/Apr/20

t h a n k y o u s i r .