Question

Form the differential equation corresponding to y² = a (b − x²) by eliminating a and b.

Answer 1

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

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

This equation contains two arbitrary constants, so we shall get a differential equation of second order.

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

\[2y\frac{dy}{dx} = - 2ax\]                              ............(2)

Differentiating equation (2) with respect to x, we get

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

From (2) and (3), we get

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

It is the required differential equation.