Question

Form the differential equation of the family of curves represented by the equation (a being the parameter):
(2x + a)² + y² = a²

Answer 1

The equation of the family of curves is \[\left( 2x + a \right)^2 + y^2 = a^2\]                                          ...(1)
where a  is a parameter.
As this equation has only one parameter, we shall get a differential equation of first order.
Differentiating (1) with respect to x, we get

\[2\left( 2x + a \right) \times 2 + 2y\frac{dy}{dx} = 0\]                                    ...(2)
Now, from (1), we get
\[4 x^2 + 4ax + a^2 + y^2 = a^2 \]
\[ \Rightarrow 4ax = - y^2 - 4 x^2 \]
\[ \Rightarrow a = - \frac{\left( 4 x^2 + y^2 \right)}{4x}\]
Putting the value of a in (2), we get
\[4\left( 2x - \frac{4 x^2 + y^2}{4x} \right) + 2y\frac{dy}{dx} = 0\]
\[ \Rightarrow 4\left( \frac{8 x^2 - 4 x^2 - y^2}{4x} \right) + 2y\frac{dy}{dx} = 0\]
\[ \Rightarrow 4 x^2 - y^2 + 2xy\frac{dy}{dx} = 0\]
\[ \Rightarrow y^2 - 4 x^2 - 2xy\frac{dy}{dx} = 0\]
It is the required differential equation.