Solve the D.E. \(3e^x\tan y\,dx + (1 + e^x)\sec^2 y\,dy = 0\).
Q.24 Solve the D.E. \(3e^x\tan y\,dx + (1 + e^x)\sec^2 y\,dy = 0\).
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Answer: \((1 + e^x)^3\tan y = c\)
Separating the variables (dividing by \((1 + e^x)\tan y\)): \(\dfrac{3e^x}{1 + e^x}dx + \dfrac{\sec^2 y}{\tan y}dy = 0\).
Integrating: \(3\displaystyle\int\frac{e^x}{1 + e^x}dx + \int\frac{\sec^2 y}{\tan y}dy = \log c\)
\(3\log|1 + e^x| + \log|\tan y| = \log c\)
\(\log|(1 + e^x)^3\tan y| = \log c\), so \((1 + e^x)^3\tan y = c\) is the general solution.
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· 28 Oct 2026
AI-verified · Oct 2026
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