You could solve the DE...
I don't believe Prove It's proof is valid. He appears to be assuming that a is a real valued function when it should be vector valued. Also, he is trying to prove the converse of the statement- that if the dot product is 0 then the curve lies in a sphere.
In fact, I don't believe the theorem, as stated, is true. Consider the curve . At any point on that curve, so the curve lies on the surface of the sphere with center at (0, 0, R) and radius R. But .
What is true is that if is a curve lieing on the surface of a sphere with center at the origin, then .
There are two ways to prove that
Analytically: Any curve lieing on the surface of a sphere with radius R and center at the origin can be written as
for some functions and
Take the dot product of those two vectors.
Geometrically: , the position vector, for each t, is a vector from the origin, the center of the sphere, to the sphere. It is, of course, perpendicular to the tangent plane at that point. And , the tangent vector to a curve lieing in the surface of the sphere, is itself in the tangent plane.