Differential Equations involving exponentials

Your second last line should be $$-e^{-y}=\frac{-1}2x+c$$ since $$\int e^{-y}dy=-e^{-y}$$


You have made an error in evaluating $$\int e^{-y(x)}dy(x)$$

The negative sign does not go away.

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Ordinary Differential Equations

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