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The general solution of the differential equation \(\left(1+e^x\right) d y+y e^x d x=0\), where \(y>0\), is
- A \(y=C\left(1+e^x\right), C\) is an arbitrary constant
- B \(\frac{c}{e^x}=\frac{y}{1+e^x}, C\) is an arbitrary constant
- C \(C=y\left(1+e^x\right), C\) is an arbitrary constant
- D \(y=\frac{C e^x}{1+e^x}, C\) is an arbitrary constant
Answer & Solution
Correct Answer
(C) \(C=y\left(1+e^x\right), C\) is an arbitrary constant
Step-by-step Solution
Detailed explanation
\(\frac{dy}{y} = -\frac{e^x}{1+e^x} dx\) \(\int \frac{dy}{y} = -\int \frac{e^x}{1+e^x} dx\) \(\ln|y| = -\ln|1+e^x| + \ln C\) \(\ln y = \ln \left(\frac{C}{1+e^x}\right)\) \(y = \frac{C}{1+e^x}\) \(C = y(1+e^x)\)
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