Question

From the differential equation of the family of circles having centre on y-axis and radius 3 units.

Answer 1

Let the centre of the circle on y-axis be(0,b).

The differential equation of the family of circles with centre at (0,b)and radius 3 is as

follows:

x2+(y-b)2=32

$\Rightarrow$ x2+(y-b)2=9 ...(1)

Differentiating equation(1) with respect to x, we get:

2x+2(y-b).y'=0

$\Rightarrow$ (y-b).y'=-x

$\Rightarrow$ y-b=-$\frac{x}{y'}$

Substituting the value of (y-b)in equation(1),we get:

$x^2+(-\frac{x}{y'})^2=9$

$\Rightarrow x^2[1+\frac{1}{(y')^2}]=9$

$\Rightarrow$ x2((y')2+1)=9(y')2

$\Rightarrow$ (x2-9)(y')2+x2=0

This is the required differential equation.