Let C be the center of the circle which is in the first quadrant.

So, Center of circle = (a, a)

Also given that the circle touches the co-ordinate Axis.

Now, equation of the circle is

     (x - a)² + (y - a)² = a²   .............1

=> x² - 2ax + a² + y² - 2ay + a² = a²

=> x² + y² - 2ax - 2ay + a² = 0

Differentiate w.r.t. x on both side, we get

=> 2x + 2y(dy/dx) - 2a - 2a(dy/dx) = 0

=> x + y(dy/dx) = a + a(dy/dx)

=> x + y(dy/dx) = a{1 + dy/dx}

=> a = {x + y(dy/dx)}/(1 + dy/dx)

Put value of a in equation 1, we get

=> [x - {x + y(dy/dx)}/(1 + dy/dx)]² + [y - {x + y(dy/dx)}/(1 + dy/dx)]² = [{x + y(dy/dx)}/(1 + dy/dx)]²   

=> [{x(1 + dy/dx) - {x + y(dy/dx)}}/(1 + dy/dx)]² + [{y(1 + dy/dx) - {x + y(dy/dx)}}/(1 + dy/dx)]² = [{x +

      y(dy/dx)}/(1 + dy/dx)]²

=> [x(1 + dy/dx) - {x + y(dy/dx)}]² + [y(1 + dy/dx) - {x + y(dy/dx)}]² = {x + y(dy/dx)}² 

=> [x + x(dy/dx) - x - y(dy/dx)]² + [y + y(dy/dx) - x - y(dy/dx)]² = {x + y(dy/dx)}²

=> [x(dy/dx) - y(dy/dx)]² + (y - x)² = {x + y(dy/dx)}²

=> (x - y)² * (dy/dx)² + (x - y)² = {x + y(dy/dx)}²

=> (x - y)² * {1 + (dy/dx)² } = {x + y(dy/dx)}²

=> {x + y(dy/dx)}² = (x - y)² * {1 + (dy/dx)² }

This is the required differential equation.