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I am taking an Engineering Math class. When it comes of differential equations with $y^2$ included in the question I am lost on the procedure used to solve it. Bernoulli method does work because its not in form required and homogeneous equations doesn't work because of different degrees.

These are a few that had me stumped (Solving any one will be appreciated.)

  1. $y' = (y^2 +9)/(xy+3y)$
  2. $y \cdot \ln(x)y' = (y^2 +36) /x$
  3. $y'=(x^2 y^4 - x^2) / y^3$

Thanks to anyone out there who can help.

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    $\begingroup$ All these equations fall in the "separable differential equations" category. You can read about that in your text. $\endgroup$ – Maesumi Sep 14 '14 at 18:13
  • $\begingroup$ Some examples $\endgroup$ – Maesumi Sep 14 '14 at 18:17
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For number one, here's a basic concept to help you get started:

$$y' = \frac{y^2 + 9}{xy + 3y} $$

$$y' = \frac{y^2 + 9}{y(x + 3)} $$

NOTE THE SEPARATION

$$\frac{y}{y^2+9}*\frac{dy}{dx} = \frac{y^2 + 9}{y(x + 3)}*\frac{y}{y^2+9}$$

$$\int \frac{y}{y^2+9}dy = \int \frac{1}{x+3} dx$$

$$\frac{\ln\left|{y^2+9}\right|}{2} = \ln\left|x+3\right| + C$$

$$\ln\left|{y^2+9}\right| = 2\ln\left|x+3\right| + C$$

$$y^2 + 9 =C(x+3)^2$$

Can you use this technique to solve the other problems? Comment if you have questions.

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  • $\begingroup$ Thank you very much. I did this in my notes but I didn't know how to integrate from there. I had arctan on the left which I see now was wrong. $\endgroup$ – Jay Sep 14 '14 at 18:29
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HINT 1

Start with the first one. Multiply across to get $(xy+3y)y'=y^2+9$. Now notice that the lhs has a factor $y$ and that $y y'$ is what you get if you differentiate $y^2$. What does that suggest?

Now look at the second one. Again we have a $y y'$. What happens if we try the same trick again?

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  • $\begingroup$ @Jay. You must have noticed that you can integrate $\frac{y^3}{y^4-1}$. Well done! But sadly there are endless tricks needed for differential equations. I am fairly hopeless. Unless you do a lot of this kind of thing, once you have passed exams etc it is much easier to use Mathematica or Maple or whatever. $\endgroup$ – almagest Sep 14 '14 at 18:38

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