Synthetic solution manifolds for differential equations (Q1818634)

By generalizing the definition of a manifold along the lines of synthetic differential geometry, the author regards the space of all solutions for a first-order differential equation as a manifold in two different ways. In contrast to the standard situation, he can prove, in this synthetic context, that given a differential equation \(y'= F(x,y)\) and real numbers \(a,b\), there exists a solution \(f:\mathbb{R}\to\mathbb{R}\) satisfying the equation with \(f(a)=b\).