Analyzing the Geometry of the Family
Imagine standing on a coordinate plane, looking at a line defined by y=x. This line, cutting through the origin at a perfect 45∘ angle, is the home for the centers of our family of circles.
Every circle in this family has its center at some point (a,a) on this line, and each has its own radius r. The equation for any such circle is given by:
This equation is the DNA of our family. It contains two parameters, a and r, which define the specific circle. Our mission is to find a differential equation that describes this entire family, independent of these specific parameters.
The Calculus of Elimination
To eliminate a and r, we turn to the power of calculus. We differentiate the circle equation with respect to x.
The derivative of (x−a)2 is 2(x−a), and the derivative of (y−a)2 is 2(y−a)y′. Since r2 is a constant, its derivative is zero. We get:
Dividing by 2, we arrive at our first-order relation:
We still have the constant a lingering. To banish it, we differentiate once more with respect to x. Applying the product rule to the second term, we get:
This is our second-order relation. Now, the stage is set for some algebraic alchemy.
The Algebraic Alchemy
We need to isolate a. From our second-order relation, we can write:
And from our first-order relation, we know:
Now, here is the brilliant trick. We want to eliminate a completely. Notice that:
Substituting our expressions for (y−a) and (x−a), we get:
y−x=(y−a)−(−(y−a)y′)=(y−a)(1+y′)
Substituting the expression for (y−a) into this, we find:
This is the moment of triumph! The constant a has vanished, leaving us with a beautiful relationship between x,y, and the derivatives.
The Final Form
Rearranging our result, we get:
(y−x)y′′=−(1+y′)(1+(y′)2)
Expanding the right side:
(y−x)y′′=−(1+(y′)2+y′+(y′)3)
Bringing everything to one side:
(y−x)y′′+(y′)3+(y′)2+y′+1=0
To match the template Py′′+Qy′+1=0, we factor out y′ from the middle terms:
(y−x)y′′+y′((y′)2+y′+1)+1=0
Comparing this to the template, we identify P=y−x and Q=(y′)2+y′+1. We have successfully navigated the geometry and the calculus to reveal the underlying structure of this family of circles.
