find out differential equation of family of circles represented by(x-h)^2+(y-k)^2=a^2

**Vinod** · Nov 25th 2011, 05:53 AM

Differentiating w.r.t.x we have

2x-2+2y dy/dx-2dy/dx =0 So,x - 1+ydy/dx-dy/dx =0

So x-1+(y-1)dy/dx=0

dy/dx=1-x/y-1

differentiating again w.r.t.x,we get

1+(y-1)d²y/dx² +(dy/dx)^2=0

Now how to calculate required differential equation.

If my calculations of 1st and 2nd order derivative is wrong, reply me accordingly.

**FernandoRevilla** · Nov 25th 2011, 06:17 AM

Differentiating both sides of $(x-h)^2+(y-k)^2=a^2$ we get $2(x-h)+2(y-k)y'=0$ or equivalently $\boxed{(x-h)dx+(y-k)dy=0}$ (this is the solution)

**Vinod** · Nov 25th 2011, 06:47 AM

Originally Posted by FernandoRevilla

Differentiating both sides of $(x-h)^2+(y-k)^2=a^2$ we get $2(x-h)+2(y-k)y'=0$ or equivalently $\boxed{(x-h)dx+(y-k)dy=0}$ (this is the solution)

Sir,
I want the solution without arbitrary constants h,k. So I want to eliminate them by differentiation.
Okay,

**FernandoRevilla** · Nov 25th 2011, 07:00 AM

Originally Posted by Vinod

Sir, I want the solution without arbitrary constants h,k. So I want to eliminate them by differentiation. Okay,

Why do you know that $h,k$ are arbitrary constants? Does the problem mention it? If this is the case, the problem should be stated: Find the differential equation of all circles in the plane.

**Vinod** · Nov 26th 2011, 04:56 AM

Originally Posted by FernandoRevilla

Why do you know that $h,k$ are arbitrary constants? Does the problem mention it? If this is the case, the problem should be stated: Find the differential equation of all circles in the plane.

Sir,
You are correct. Original problem is as follows:
Find the differential equation of all circles having a fixed radius a.

Answer given in the study material is as below:
{1+ (dy/dx)^2}^3=a^2{d²y/dx²}^2

Hence give me a hint to arrive at the above answer.

Regards

**Vinod** · Dec 6th 2011, 03:43 AM

Sir,
I found the differential equation of all circles having fixed radius a.First derivative of equation of these circles is
$\frac{dy}{dx}=\frac{-(x-h)}{(y-k)}$

Second derivative of equation of these circles is
d²y/dx²=[-(dy/dx)^2-1]/(y-k)

Differential equation of all circles having fixed radius a is given in post number #5 to this thread.

However thanks for your reply.

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