I need some explanation regarding a solution found in our textbook.(adsbygoogle = window.adsbygoogle || []).push({});

The example starts with this DE:

(1) y(x^{3}- y)dx - x(x^{3}+ y)dy = 0.

By regrouping, exact DEs will be found and the equation can be rewritten to:

(2) x^{3}d([tex]\frac{x}{y}[/tex]) - [tex]\frac{d(xy)}{y}[/tex] = 0.

The integrating factor is found to be: x^{-2}y^{-1}. Multiplying the equation by the IF results to:

(3) [tex]\frac{x}{y}[/tex] d([tex]\frac{x}{y}[/tex]) - [tex]\frac{d(xy)}{x^{2}y^{2}}[/tex]

Now this is the part that I don't understand.The text continues after the equation above, "of which a set of solutions is given by"

(4) [tex]\frac{1}{2}[/tex] ([tex]\frac{x}{y}[/tex])^{2}+ [tex]\frac{1}{xy}[/tex] = [tex]\frac{c}{2}[/tex].

Just how did (3) turn into (4)?

I think it's a very simple question but I'm really lost. I'm so confused with anything that has to do with differential equations. :( Thanks in advance for the help.

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