Diff Eq problem

  • Thread starter Maxwell
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  • #1
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Here is the problem:

Determine the orthogonal trajectories of the given family of curves.

[tex] y = \sqrt{2\ln{|x|}+C} [/tex]

This is what I've done so far:

[tex] y = (2\ln{|x|}+C)^\frac{-1}{2} [/tex]

[tex] y' = -1/2(2\ln{|x|+C)(2/x) [/tex]

Now I understand to find the orthogonal lines I need to divide -1 by whatever I get, the problem is, I can't simplify this derivative.

I've messed around with it a bit, and I have this:

[tex] -(2\ln{|x|}+C)/x [/tex]

How else can I simplify this?

Thanks.
 

Answers and Replies

  • #2
Tide
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First, your derivative is incorrect. Your final result may be written as

[tex]y' = \frac {1}{xy}[/tex]

Does that help?
 
Last edited:
  • #3
ehild
Homework Helper
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Maxwell said:
Here is the problem:

Determine the orthogonal trajectories of the given family of curves.

[tex] y = \sqrt{2\ln{|x|}+C} [/tex]

This is what I've done so far:

[tex] y = (2\ln{|x|}+C)^\frac{-1}{2} [/tex]

Why is the power negative?

There is an easier way. Just square the original equation, and differentiate with respect to y.

[tex]y^2=2\ln{|x|}+C[/tex]

[tex]2yy'=\frac{2}{x}[/tex].
....
express y', take the negative reciprocal and you get the differential equation for the trajectories. Solve, it is easy.

ehild
 

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