Discussion
The integrating factor of the differential equation \((1 + x^{2})dy = (1 - 2xy)dx\) is
Answer: A
The integrating factor is \(x^{2} + 1\).
Put the equation in linear form:
\[ \frac{dy}{dx} + \frac{2x}{1 + x^{2}}y = \frac{1}{1 + x^{2}} \]
The integrating factor is \(e^{\int P\,dx} = e^{\log(1 + x^{2})} = 1 + x^{2}\). \(\log_{e}(x^{2} + 1)\) stops before taking the exponential, which is the usual slip; the other options do not come from \(\int\frac{2x}{1 + x^{2}}dx\).
Multiplying through by \(1 + x^{2}\) gives \(\frac{d}{dx}[(1 + x^{2})y] = 1\), so the solution is \((1 + x^{2})y = x + C\).
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