xdy-[y + xy^2 ( 1+ ln x)] = dx

complete solution with final answer

To solve the given differential equation:
\[
x \, dy - \left[y + xy^2 \left(1 + \ln{x}\right)\right] = dx
\]
we will follow these steps:

1. **Rearrange the equation.**

   Rewrite the equation in the form:
   \[
x \, dy - dx = \left[y + xy^2 \left(1 + \ln{x}\right)\right]
   \]

2.

xdy - [y + xy^2 (1 + ln x)] = dx. Complete solution with final answer.

This page provides a complete solution to the differential equation xdy - [y + xy^2 (1 + ln x)] = dx, covering key concepts in calculus and differential equations. Follow a detailed, step-by-step breakdown, including logarithmic differentiation and integration techniques.

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