Solution of the differential equation e^x-e^ -x e^x+e^ -x =d x-d … - Vidyadip

Question

Solution of the differential equation $\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{d x-d y}{d x+d y}$, is

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Answer

$(b):$ Given equation $\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{d x-d y}{d x+d y}$
Applying componendo and dividendo, we get
$\frac{d y}{d x}=\frac{e^{-x}}{e^x}$
$\Rightarrow d y=e^{-2 x} d x$
$\Rightarrow 2 y=-e^{-2 x}+C ($Integrating both sides$)$
$\Rightarrow 2 y e^{2 x}=C \cdot e^{2 x}-1$

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