<s>[INST] First Order Separable differential Equation 
Problem:
  Solve the following differential equation:
  \begin{eqnarray*}
6x^2y \, dx - (x^3 + 1) \, dy &=& 0 \\
\end{eqnarray*}

Answer:
This is a separable differential equation.
\begin{eqnarray*}
\frac{6x^2}{x^3+1} \, dx - \frac{dy}{y} &=& 0 \\
\int \frac{6x^2}{x^3+1} \, dx - \int \frac{dy}{y} &=& c_1 \\
2 \ln{|x^3+1|} - \ln{|y|} &=& c_1 \\
\ln{(x^3+1)^2} - \ln{|y|} &=& c_1 \\
\ln{ \Big( \frac{(x^3+1)^2}{|y|} \Big) } &=& c_1 \\
(x^3+1)^2 &=& c|y| \\
\end{eqnarray*}
However, the book gets:
\begin{eqnarray*}
(x^3+1)^2 &=& |cy| \\
\end{eqnarray*}
Is my answer different from the book's answer? I believe it is. What am I missing?Any idea of how to proceed?
Thanks,
Bob
 [/INST] Both answers are correct.
Your answer $$(x^3+1)^2 = c |y|$$ makes the assumption that $c\ge 0$
The book's answer $$(x^3+1)^2 = |cy|$$ is OK for all values of $c$.
Thus to make sure that you can take any   value for c go with the book's answer, otherwise mention that $c\ge 0$ .  
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