Question

Form the differential equation corresponding to (x − a)² + (y − b)² = r² by eliminating a and b.

Answer 1

The equation of the family of curves is \[\left( x - a \right)^2 + \left( y - b \right)^2 = r^2...............(1)\]

where \[a\text{ and }b\] are parameters.

This equation contains two parameters, so we shall get a second order differential equation.

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

\[2\left( x - a \right) + 2\left( y - b \right)\frac{dy}{dx} = 0...............(2)\]

Differentiating (2) with respect to x, we get

\[2 + 2 \left( \frac{dy}{dx} \right)^2 + 2\left( y - b \right)\frac{d^2 y}{d x^2} = 0\]

\[ \Rightarrow 1 + \left( \frac{dy}{dx} \right)^2 + \left( y - b \right)\frac{d^2 y}{d x^2} = 0\]

\[ \Rightarrow \left( y - b \right) = - \frac{1 + \left( \frac{dy}{dx} \right)^2}{\frac{d^2 y}{d x^2}} .................(3)\]

From (2) and (3), we get

\[\left( x - a \right) - \frac{1 + \left( \frac{dy}{dx} \right)^2}{\frac{d^2 y}{d x^2}}\frac{dy}{dx} = 0\]

\[ \Rightarrow \left( x - a \right) = \frac{\frac{dy}{dx} + \left( \frac{dy}{dx} \right)^3}{\frac{d^2 y}{d x^2}} .................(4)\]

From (1), (3) and (4), we get 

\[\frac{\left[ \frac{dy}{dx} + \left( \frac{dy}{dx} \right)^3 \right]^2}{\left( \frac{d^2 y}{d x^2} \right)^2} + \frac{\left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^2}{\left( \frac{d^2 y}{d x^2} \right)^2} = r^2 \]
\[ \Rightarrow \frac{\left[ \left( \frac{dy}{dx} \right)^2 + 2 \left( \frac{dy}{dx} \right)^4 + \left( \frac{dy}{dx} \right)^6 \right] + \left[ 1 + 2 \left( \frac{dy}{dx} \right)^2 + \left( \frac{dy}{dx} \right)^4 \right]}{\left( \frac{d^2 y}{d x^2} \right)^2} = r^2 \]
\[ \Rightarrow \left( \frac{dy}{dx} \right)^2 + 2 \left( \frac{dy}{dx} \right)^4 + \left( \frac{dy}{dx} \right)^6 + 1 + 2 \left( \frac{dy}{dx} \right)^2 + \left( \frac{dy}{dx} \right)^4 = r^2 \left( \frac{d^2 y}{d x^2} \right)^2 \]
\[ \Rightarrow 1 + 3 \left( \frac{dy}{dx} \right)^2 + 3 \left( \frac{dy}{dx} \right)^4 + \left( \frac{dy}{dx} \right)^6 = r^2 \left( \frac{d^2 y}{d x^2} \right)^2 \]
\[ \Rightarrow \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = r^2 \left( \frac{d^2 y}{d x^2} \right)^2 \]
It is the required differential equation.