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Which of the following represents the solution of the differential equation
dy/dx + (y/x) log y = (y/x²) (log y)² ?
- y = log(x² + cx)
- log y = x(cx² + 1/2)
- x = log y(cx² + 1/2)
- None of these
Correct answer: x = log y(cx² + 1/2)
Solution
Put v=ln y: v' + v/x = v^2/x^2 (Bernoulli). With w=1/v: w' - w/x = -1/x^2, giving w = c*x + 1/(2x), i.e. 1/ln y = c*x + 1/(2x). Rearranged this is x = ln y*(c x^2 + 1/2), which satisfies the ODE (option 2).
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