SOLUTION: Find the differential equation representing all circles with center on the line y = x and passing through the origin. Draw some representation of the family

Question 1186122: Find the differential equation representing all circles with center on the line y = x and passing
through the origin. Draw some representation of the family
Found 2 solutions by ikleyn, robertb:
Answer by ikleyn(52790) (Show Source): You can put this solution on YOUR website!
.

The term "differential equation", which you use in your post, is INCORRECT and IRRELEVANT in this context.

The correct formulation is

Find the standard form equation representing all circles with center on the line y = x and passing through the origin.

Ask your professor/teacher or double check with your source, then re-post the corrected version to the forum.

Please do not post it to me personally.

Also, do not ask me to make drawings for you.
At this forum, we are not an art studio to make drawings on request of visitors.
All you need to make the drawing on your own, is a pen, page of paper, a straightedge and a compass.

Do not forget to post your "THANKS" to me for pointing deficiency in your post.

Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!

The equation of the circle with center at (h,k) and having radius r is given by

Since the center is on the line y = x, then h = k, hence we have
.

Since every circle in the family of circles passes through the point (0,0), we get
, which implies that the equation of the family of circles is
.

Differentiating with respect to x while treating h as a parameter, we get

====> , or .

Now expand the equation for the family of circles. This gives , which leads to
, or , or .
Substituting for h into the equation above (in red ink), we obtain

.
Now solve for .
, or

.

Simplifying, we obtain

, or finally, ,

and that is the differential equation representing all circles with center on the line y = x and passing through the origin.

I leave the drawing part up to you. You should be capable enough of doing that, if you were qualified enough to take that course on ODE's.

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