If the differential equation d y d x + x y = a y where a is constant, represents a family of circles then the radius of the circle is ......

Step-by-step solution

The differential equation dy dx + x y = a y can be transformed by multiplying both sides by y to eliminate denominators, yielding y dy dx + x = a . Rearranging terms separates the variables as y ,dy = (a - x) ,dx . Integrating both sides gives y^2 2 = ax - x^2 2 + C₁ , where C₁ is the constant of integration. Multiplying through by 2 produces y^2 = 2ax - x^2 + 2C₁ . Completing the square for the x -terms by adding a^2 to both sides results in (x - a)^2 + y^2 = a^2 - 2C₁ . This represents a circle centered at (a, 0)