Question

the order of the differential equation of all circles having radius r is

Answer 1

The General equation of circle is

(x-a)² +(y-b)²= r²-------(A)

There are two constant in this equation of circle.We can differentiate it twice only to remove these variables

Differentiating With respect to x

→2 (x-a) + 2 (y-b)× y'=0

→ x-a + y y' - b y'=0

→x-a= -y'(y-b)------(1)

Differentiating again with respect to x

→→1 + (y')²+y y"-b y"=0

→y"(y-b)= -[ 1 +(y')²]

→y-b= $\frac{-( 1 +(y')^2)}{y"}$

Equation (1) becomes

x-a= -y'× $\frac{-( 1 +(y')^2)}{y"}$

Substituting the values of (x-a) and (y-b) in equation (A)

[-y'× $\frac{-( 1 +(y')^2)}{y"}$]²+[ $\frac{-( 1 +(y')^2)}{y"}$]²=r²

→ (1+y'²)²[y'²+1]=r²×y"²

→(1+y'²)³=r²×y"²

we have differentiated it twice, so it's order is 2, and power of highest order that is y" is also 2, so it's degree is also 2.