Formation of a Differential Equation Whose General Solution is Given




The equation is  `x^2 + y^2 + 2x – 4y + 4 = 0` ... (1)
represents a circle having centre at (–1, 2) and radius 1 unit.
Differentiating equation (1) with respect to x, we get
`(dy)/(dx) = (x + 1)/(2 - y)` (y ≠ 2)    ...(2)
which is a differential equation. You will find later on  that this equation represents the family of circles and one member of the family is the circle given in equation (1). 
Let us consider the equation:
`x^2 + y^2 = r^2 `... (3) 
By giving different values to r, we get different members of the family e.g. `x^2 + y^2 = 1`, `x^2 + y^2 = 4`, `x^2 + y^2 = 9` etc   Fig .

Thus, equation (3) represents a family of concentric circles centered at the origin and having different radii. 
We are interested in finding a differential equation that is satisfied by each member of the family. The differential equation must be free from r because r is different for different members of the family. This equation is obtained by differentiating equation (3) with respect to x, i.e.,
2x + 2y `(dy)/(dx)` = 0 or  x + y `(dy)/(dx)` =0   ...(4)
which represents the family of concentric circles given by equation (3). 

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