The general solution of the differential equa | vedclass

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(C) Given the differential equation: $\frac{dy}{dx} = e^{x+y}$.Using the property of exponents,we can write: $\frac{dy}{dx} = e^x \cdot e^y$.Separatin

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$e^{x} + e^{-y} = C$

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(C) Given the differential equation: $\frac{dy}{dx} = e^{x+y}$.Using the property of exponents,we can write: $\frac{dy}{dx} = e^x \cdot e^y$.Separating the variables,we get: $\frac{dy}{e^y} = e^x dx$,which is equivalent to $e^{-y} dy = e^x dx$.Integrating both sides: $\int e^{-y} dy = \int e^x dx$.This yields: $-e^{-y} = e^x + k$,where $k$ is the constant of integration.Rearranging the terms: $e^x + e^{-y} = -k$.Letting $C = -k$,we get the general solution: $e^x + e^{-y} = C$.Thus,the correct option is $C$.

The general solution of the differential equation $\frac{dy}{dx} = e^{x+y}$ is

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