- #1

SU403RUNFAST

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## 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?