Differential Equations

[SuzyGold]

Hi guys , its my first time here and I really need help solving this problem: Form the differential equation of the family of all the circles, with center (m,n) and passing through the origin. I would really appreciate any replies. Please and thank you.

 

[lillybeth]

Was there an illustration with the problem??

 

[Ishuda]

Hi guys , its my first time here and I really need help solving this problem: Form the differential equation of the family of all the circles, with center (m,n) and passing through the origin. I would really appreciate any replies. Please and thank you.

How far is it from the origin at (0,0) to the point (m,n) (the center of the circle)? If a point [(0,0)] lies on a circle, how far is it from the center [ (m,n) ]? What is the formula of a circle with center at (m,n) and radius r?

Now do an implicit differentiation of the equation to get a value for the derivative of y. Then substitute back for y.

 

[HallsofIvy]

Since there are two parameters, m and n, you will want to differentiate twice to eliminate both parameters. That will give you a second order differential equation whose solution has two constants, the m and n.

 

[Ishuda]

The family of circles with center (n,m) going through the origin is defined as
(x-n)² + (y-m)² = n² + m²

or
(x-n) + (y-m) y' = 0; y(0) = 0
or go one further
(y-m) y'' + (y')² = -1; y(0) 0, y'(0) = -n/m
or ...