Ind the orthogonal trajectories of the family of curves

[Bel_Oubli]

Homework Statement

Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen.
y = x/(1+kx)

2. The attempt at a solution

I have been trying this problem for hours, and I get a different answer every time. I have a big exam over this material tomorrow, and I just discovered that someone in this forum may be able to help me. Any help will be GREATLY appreciated!

The following is my most recent attempt. This was after I got help from a friend. I know that it must be wrong, but it's as far as I have gotten.

y = x/(1+kx)
y' = dy/dx = x/[(x+kx)^2]
slope of tangent line = x/[(x+kx)^2]
slope of orthogonal line = -(1+kx)^2
dy = [-(1+kx)^2]dx

After that, I am completely stuck. I do not think I am anywhere close to the answer. Also, I have no idea how to go about graphing it. Please help!

 

[Dick]

I'm ignoring what I think are typos, like the correct y' is 1/(1+kx)^2. Your final line looks good though. Now just integrate both sides. Use a substitution u=1+kx. Or just square it out if you have to.

 

[Bel_Oubli]

Do you really think I am on the right track? I thought I was supposed to take K out early on, but I do not know when, or what to replace it with. I just think that my final answer is not supposed to have K in it.

 

[Dick]

Do you really think I am on the right track? I thought I was supposed to take K out early on, but I do not know when, or what to replace it with. I just think that my final answer is not supposed to have K in it.

Yes, I think you are on the right track. And I don't see why you think the final answer shouldn't have a k in it.

 

[Bel_Oubli]

My final answer is y=-[(1+KX)^3]/3K + C. I think that should be correct. To graph it, will I just graph the original equation y = x/(1+kx) and then add y=-[(1+KX)^3]/3K + C for different values of K and C?

In my math book, it says that K is an arbitrary constant and that you have to do it for all values of K, so you find K as follows:
y = x/(1+kx)
1+kx = x/y
k=(x-1)/y

I just do not know when I would add it into the problem. Perhaps this is correct:
y = x/(1+kx)
y' = dy/dx = x/[(x+kx)^2]
And replacing k with (x-1)/y...
slope of tangent line = x/[(x+[(x-1)/y]x)^2] = (xy^2)/(x^4)

Does that look right? I'm so confused...

 

[Dick]

For someone who has done all the right stuff, you are mighty confused. Let f(x)=x/(1+kx) and g(x)=c+(1+kx)^3/(-3k). You have shown that f'(x)=-1/g'(x). So if you put in any values of c and k, then if f(x) and g(x) intersect, they will intersect at a right angle. They are orthogonal.

 

[Bel_Oubli]

LOL, I think I'm just getting stressed out because it's almost the end of the semester, and I want to be prepared for Calculus III next semester. Thank you so much for helping me; I will just go to bed and hope I pass the exam tomorrow.

Thanks again!
(:

 

[Dick]

Good strategy, you probably will if you get some sleep. You are doing a lot of stuff right.

 

[HallsofIvy]

Dick, the differential equation defining the family y= x/(1+kx) should NOT have a "k" in it. The k determines which member of the family you have. Once you have determined y'= f(x,y), write y'= -1/f(x,y) and integrate that to find the orthogonal trajectories. You will get a constant of integration in that, of course.

I would be inclined to do it this way: write y= x/(1+kx) as y(1+ kx)= x. That allows you to do two things. First, differentiating both sides, y'(1+ kx)+ ky= 1. Also y+ kxy= x so kxy= x- y and you have both kx= (x-y)/y and ky= (x-y)/x. y'(1+ (x-y)/y)+ (x-y)/x= 1.
1+ (x-y)/y= x/y and 1- (x-y)/x= y/x. The equation of the original family is (x/y)y'= (y/x) or y'= y²/x².

 

[Dick]

Thanks, Halls. Or just write (1+kx)=x/y and put that into the derivative the OP already has y'=1/(1+kx)=y^2/x^2. I had a feeling something was going awry...

 

[Bel_Oubli]

Wow, my answer was really wrong. I guess it does not matter since my exam has already passed, but thanks for the help, guys.