Question

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y² = 4a (x − b)

 

Answer 1

The equation of family of curves is \[y^2 = 4a\left( x - b \right)\]                                                ...(1)
where \[a\text{ and }b\] are parameters.
As this equation has two arbitrary constants, we shall get a differential equation of second order.
Differentiating (1) with respect to x, we get
\[2y\frac{dy}{dx} = 4a\]
\[ \Rightarrow y\frac{dy}{dx} = 2a . . . \left( 2 \right)\]
Differentiating (2) with respect to x, we get
\[y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 = 0\]
It is the required differential equation.