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 $a\mathrm{and}b$ 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
$$\begin{gathered} 2+2{\left(\frac{dy}{dx}\right)}^2+2\left(y-b\right)\frac{d^{2}y}{d{x}^2}=0 \\ \Rightarrow 1+{\left(\frac{dy}{dx}\right)}^2+\left(y-b\right)\frac{d^{2}y}{d{x}^2}=0 \\ \Rightarrow \left(y-b\right)=-\frac{1+{\left({\displaystyle \frac{dy}{dx}}\right)}^2}{{\displaystyle \frac{d^{2}y}{d{x}^2}}}...\text{(3)} \end{gathered}$$
From (2) and (3), we get
$$\begin{gathered} \left(x-a\right)-\frac{1+{\left({\displaystyle \frac{dy}{dx}}\right)}^2}{{\displaystyle \frac{d^{2}y}{d{x}^2}}}\frac{dy}{dx}=0 \\ \Rightarrow \left(x-a\right)=\frac{{\displaystyle \frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^3}}{{\displaystyle \frac{d^{2}y}{d{x}^2}}}...\text{(4)} \end{gathered}$$
From (1), (3) and (4), we get
$$\begin{gathered} \frac{{\left[{\displaystyle \frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^3}\right]}^2}{{\left({\displaystyle \frac{d^{2}y}{d{x}^2}}\right)}^2}+\frac{{\left[1+{\left({\displaystyle \frac{dy}{dx}}\right)}^2\right]}^2}{{\left({\displaystyle \frac{d^{2}y}{d{x}^2}}\right)}^2}=r^2 \\ \Rightarrow \frac{\left[{\left({\displaystyle \frac{dy}{dx}}\right)}^2+2{\left({\displaystyle \frac{dy}{dx}}\right)}^4+{\left({\displaystyle \frac{dy}{dx}}\right)}^6\right]+\left[1+2{\left({\displaystyle \frac{dy}{dx}}\right)}^2+{\left({\displaystyle \frac{dy}{dx}}\right)}^4\right]}{{\left({\displaystyle \frac{d^{2}y}{d{x}^2}}\right)}^2}=r^2 \\ \Rightarrow {\left(\frac{dy}{dx}\right)}^2+2{\left(\frac{dy}{dx}\right)}^4+{\left(\frac{dy}{dx}\right)}^6+1+2{\left(\frac{dy}{dx}\right)}^2+{\left(\frac{dy}{dx}\right)}^4=r^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2 \\ \Rightarrow 1+3{\left(\frac{dy}{dx}\right)}^2+3{\left(\frac{dy}{dx}\right)}^4+{\left(\frac{dy}{dx}\right)}^6=r^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2 \\ \Rightarrow {\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^3=r^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2 \\ \text{It is the required differential equation.} \end{gathered}$$