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