Differential equations, orthogonal trajectories

[SU403RUNFAST]

Homework Statement

you are given a family of curves, in this case i was given a bunch of circles x^2+y^2=cx, sketch these curves for c=0,2,4,6, both positive and negative, solve the equation for c and differentiate both sides with respect to x and solve for dy/dx. You obtain an ODE in the form of dy/dx=f(x,y), now any orthogonal or perpendicular trajectory must be perpendicular at any point, so we're looking for curves satisfying dy/dx=-1/f(x,y) (negative reciprocal slope). Find an implicit solution to this ODE, and sketch these

Homework Equations

x^2+y^2=cx

The Attempt at a Solution

I plotted all the circles, the first one c=0 is a point, and the rest are circles on the left and right sides of the origin, each new circle's radius moves outwards by 1. This was easy part, and solving for c i got c=(x^2+y^2)/x, i split this up into c=x+y^2/x and differentiated to get 0=1-(y^2/x^2)(dy/dx) so that means dy/dx=x^2/y^2. I took the reciprocal so the new ODE becomes dy/dx=-y^2/x^2, but the problem is that when i solve this ODE i get y=Ce^(1/x) and when i plot this for random C values none of these are perpendicular to my circles! I would like to know where my mistake is, did I differentiate in the first step correctly? Where did I go wrong?

 

[Noctisdark]

This is the equation if a circle x^2 + y^2 = K,
So what you have above is a circle just rearrange to check .
Differentiation gives 2*x + 2*y*y' = c
Solve for dy/dx = (c-2x)/(2*y)
What you need is dy/dx = -(2*y)/(c-2x)
This is a seperable 1 ODE that can be solve by re-arranging to
(1/2*y)dy = dx/(c-2x)
And it's up to you ;) Good luck

 

[SU403RUNFAST]

okay i am going to use your work and solve and see if it is perpendicular thanks, but i plotted the original solutions and they came out as circles,i rewrote them as (x-c/2)^2+y^2=(c/2)^2, i will check

 

[SU403RUNFAST]

i guess its the same if you differentiate it without solving for c first

 

[Noctisdark]

I diff the whole equation without re-arrangement, didn't got rid of anything, they are indeed circles,if you take a look, everything is Okay because (x^2)' = 2x , (cx)' = c , (y^2)' = 2yy'

 

[SU403RUNFAST]

Yes I understood it, i have now obtained -ln(y)/2 + C = -ln(c-2x)/2 + C i think to plot I have to choose the parameter C, thanks

 

[SU403RUNFAST]

Unless they cancel out, but i don't think they would have the same value because they are different constants

 

[SU403RUNFAST]

wait nvm i got it, but just to make sure can i just get rid of the big constant C's and then all is left is ln(y)=ln(c-2x)

 

[Noctisdark]

Ln(y) = ln(c-2x) + K (K =(C-c)/2 is the integration constants)
Y = c-2x + Cte (constant of integration)

 

[SU403RUNFAST]

why are you subtracting C-c... and where does division by 2i come from? There arent any irrational numbers... What the hell is C t and e.. i can't plot this with so many random constants

 

[Noctisdark]

Oops, keyboard typing errors i means C-c/2 = K to treat both of them as one constant C and c are constant if ODE solving

 

[SU403RUNFAST]

Little c is a number though, 0,-2,2,-4,4,etc, I am going to assume you mean C-C/2=K, so i did the same thing to simplify by raising e^ln(), are you sure you can just take an average randomly to get K or it doesn't matter since it doesn't show up, it only shows up as K. So do I pick random K values? This needs to be perpendicular to x^2+y^2=cx

 

[SU403RUNFAST]

Plotting y=c-2x alone gives lines through each circle through origin so that works if i ignore K, thanks for your immense help and clarifying my stupid questions

 

[Noctisdark]

K is an integration constant, in fact i didnt mean to write it as C-c/2 but I can so why not ? it show up as a single constant

 

[Noctisdark]

Welcome, but don't forget to write as y= c-2x + K, in your case K = 0 happened to work, but allways write the most general solution and pick up the values you need, (you picked up K= 0 and that's great )

 

[SU403RUNFAST]

wait yeah but wouldn't when i simplifed using e^()=e^() +e^K wouldn't this k=0 add a 1 and mess up the solution...

 

[Noctisdark]

e^K = Cte an other constant
After you write that, ignore K, it's all about Cte

 

[SU403RUNFAST]

So i just resolved the problem a different way and at the end I got y=(x^2+y^2)c. These are more circles that are orthogonal

 

[HallsofIvy]

So i just resolved the problem a different way and at the end I got y=(x^2+y^2)c. These are more circles that are orthogonal

"More circles that are orthogonal" to what?