Solving Orthogonal Trajectories of a Family of Curves

[Maxwell]

Here is the problem:

Determine the orthogonal trajectories of the given family of curves.

$$y = \sqrt{2\ln{|x|}+C}$$

This is what I've done so far:

$$y = (2\ln{|x|}+C)^\frac{-1}{2}$$

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

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:

$$-(2\ln{|x|}+C)/x$$

How else can I simplify this?

Thanks.

 

[Tide]

First, your derivative is incorrect. Your final result may be written as

$$y' = \frac {1}{xy}$$

Does that help?

 

[ehild]

Here is the problem:

Determine the orthogonal trajectories of the given family of curves.

$$y = \sqrt{2\ln{|x|}+C}$$

This is what I've done so far:

$$y = (2\ln{|x|}+C)^\frac{-1}{2}$$

Why is the power negative?

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

$$y^2=2\ln{|x|}+C$$

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

ehild