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.