General And Particular Solutions Of A Differential Equation

The equation of the family of circles of radius r is
...(1)
Where a & b are arbitrary constants.
Since equation (1) contains two arbitrary constants, we differentiate it two times w.r.t x & the differential equation will be of second order.
Differentiating (1) w.r.t. x, we get


Differentiating (2) w.r.t. x, we get


On putting the value of (y - b) in equation(2), we get

Substituting the values of (x - a) &(x - b)in (1), we get

The given differential equation is



This is the linear differential equation.
Put y = vx, so that Then



Integrating both sides, we get

The differential equation of the rectangular hyperbola is