# In finding a differential equation to representing something (e.g. family of conic sections), is it necessary to use integration or not?

In the question find the DE of the family of circles of fixed radius r with centers on the x-axis, I saw the answer only containing differentials. Even the method used was only differentiation.

When I tried answering this question, "form the DE representing all tangents to the parabola y^2=2x," I used differentiation and integrated afterward. Is this wrong? Should I just differentiate and get the equation from that?

A. Find the DE of the family of circles of fixed radius r with centers on the x-axis.

answer>>>>y^2[(dy/dx)^2 + 1] = r^2

I already got the answer to this.

B. Form the DE representing all tangents to the parabola y^2 = 2x.

I'm a bit confused about what method/route to be taken for this.

### 2 Answers

- CLv 52 months ago
Maybe if you typed up the questions (instead of your brief explanations of the questions), one of us could answer.

- Ian HLv 72 months ago
A: (x – c)^2 + y^2 = r^2 ...........................(1)

describes any of a family of circles of fixed radius r with centres on the x-axis.

If the question is about finding a related differential equation you have

2(x – c) + 2ydy/dx = 0, or,

dy/dx = (c – x)/y

B: Let y = mx + c be a tangent to the parabola y^2 = 2x

Upper curve locations can conveniently be represented by points P(a^2/2, a)

2ydy/dx = 2, so a tangent slope is m = dy/dx = 1/y = 1/a

a = (1/a)*(a^2/2) + c

c = a/2

y = (1/a)x + a/2 is a general form of a tangent

For example if a = 4 then P is (8, 4) and tangent is

y = x/4 + 2

https://www.wolframalpha.com/input/?i=y%5E2+%3D+2x...

Perhaps what is wanted is dy/dx = 1/y for that part.