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Given the first-order differential equation: $$\frac{dy}{dx} = -6xy$$ The textbook says you cannot differentiate both sides as $y$ is on the right side and you have to use separation of variables. However, I did integrate both sides and arrived at this: $$\int{\frac{dy}{dx}dx} = \int{-6xydx}$$ $$y = -6y\int{xdx}$$ $$y = -6y\left(\frac{x^2}{2} + C\right)$$ $$y = -3yx^2 + C$$ $$y + 3yx^2 = C$$ $$y(1 + 3x^2) = C$$ $$y(x) = \frac{C}{1+x^2}$$

And the second last step is valid since $1 + 3x^2$ can never be zero. However, this is not the correct answer, which is: $y(x) = Ce^{-3x^2}$. Why is this?

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    $\begingroup$ You cannot take $y$ off the integral, in the right part, since $y$ is a function of $x$. $\endgroup$
    – Mr. N
    Feb 12, 2020 at 1:41
  • $\begingroup$ Can't, yes? Okay, I see that, thanks very much. $\endgroup$ Feb 12, 2020 at 1:42
  • $\begingroup$ How do I mark the comment as the solution? $\endgroup$ Feb 12, 2020 at 1:42
  • $\begingroup$ @Mr.N So there is no other way to solve that integral on the right? $\endgroup$ Feb 12, 2020 at 1:44
  • $\begingroup$ It's separable separate first then integrate $\endgroup$ Feb 12, 2020 at 1:47

1 Answer 1

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Basically, you cannot take $y$ off the integral, in the right part, since $y=y(x)$. For example, if we have:

$$\displaystyle\int y(x) dx$$

If $y(x) =e^x$, then

$$ e^x \neq e^x \displaystyle\int dx $$ $$ e^x \neq e^x \cdot x $$

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  • $\begingroup$ So it is not possible to solve an integral with $y$ in it? $\endgroup$ Feb 12, 2020 at 1:45
  • $\begingroup$ No, unless you have $y$ in terms of $x$ only. Like in the example above. $\endgroup$
    – Mr. N
    Feb 12, 2020 at 1:45
  • $\begingroup$ I mean for that equation $y$ is only in terms of $x$ it is just that I don't know what the function is. $\endgroup$ Feb 12, 2020 at 1:47
  • $\begingroup$ That is it. If you knew, you could. $\endgroup$
    – Mr. N
    Feb 12, 2020 at 1:48
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    $\begingroup$ Okay, thanks. I will accept your answer when the time is done. $\endgroup$ Feb 12, 2020 at 1:50

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