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Question
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
Solution
We have
\[\frac{dy}{dx} = 2^{- y} \]
\[ \Rightarrow \frac{dy}{2^{- y}} = dx\]
\[ \Rightarrow 2^y dy = dx\]
Integrating both sides, we get
\[\int 2^y dy = \int dx\]
\[ \Rightarrow \frac{2^y}{\log2} = x + c\]
\[ \Rightarrow 2^y = x\log2 + k, \text { where } k = c\log2\]
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