What is the general solution of (1 + e^x) y dy = e^x dx ? (where c is a constant of integration) - Mathematics MCQ Mathematics Mock Test - 2 2026 | ExamTest.live

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What is the general solution of (1 + e^{x}) y d y = e^{x} d x ? (where c is a constant of integration)

Solution & Step-by-step Explanation

Given: (1 + e^{x}) y d y = e^{x} d x Separate the variables: y d y = \frac{e ^{x}}{1 + e ^{x}} d x Integrate both sides: \int y d y = \int \frac{e ^{x}}{1 + e ^{x}} d x For the RHS, let u = 1 + e^{x}, then d u = e^{x} d x : \frac{y ^{2}}{2} = ln (1 + e^{x}) + ln c \frac{y ^{2}}{2} = ln [c (1 + e^{x})] Multiply by 2: y^{2} = 2 ln [c (1 + e^{x})] y^{2} = ln [c (1 + e^{x})]^{2} y^{2} = ln [c^{2} (1 + e^{x})^{2}]

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What is the general solution of (1 + e^{x}) y d y = e^{x} d x ? (where c is a constant of integration)

y^{2} = ln [c^{2} (e^{x} + 1)^{2}]

y = ln [c (e^{x} + 1)]

y^{2} = ln [c (e^{x} + 1)]

None of the above

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