Question

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

Answer 1

The equation of the family of curves is
\[\left( x - a \right)^2 + 2 y^2 = a^2 \]
\[ \Rightarrow x^2 - 2ax + a^2 + 2 y^2 = a^2 \]
\[ \Rightarrow x^2 - 2ax + 2 y^2 = 0 . . . \left( 1 \right)\]
where a is a parameter.
As this equation has only one arbitrary constant, we shall get a differential equation of first order.
Differentiating (1) with respect to x, we get
\[2x - 2a + 4y\frac{dy}{dx} = 0\]                                       ...(2)
\[2a = \frac{x^2 + 2 y^2}{x}\]                                             ...(3)
From (2) and (3), we get
\[2x - \frac{x^2 + 2 y^2}{x} + 4y\frac{dy}{dx} = 0\]
\[ \Rightarrow 2 x^2 - x^2 - 2 y^2 + 4xy\frac{dy}{dx} = 0\]
\[ \Rightarrow 4xy\frac{dy}{dx} + x^2 - 2 y^2 = 0\]
\[ \Rightarrow 4xy\frac{dy}{dx} = 2 y^2 - x^2 \]
It is the required differential equation.