# Mathematical applications of ordinary differential equations.

I'm looking for more mathematically oriented applications of ODEs (if possible of first order equations). I've browsed through several books and they are all full of physics applications and very (very!) few mathematical applications. I already know about finding an orthogonal curve to a family of curves. Could you point out some applications or if possible a book were I could find some?

Although I'm familiar with differential geometry I'd rather have applications that are more calculus flavored.

Prove functional identities by checking both sides satisfy the same ODE with the same initial conditions. For instance, $e^{a+x} = e^ae^x$ since both sides satisfy $f'(x) = f(x)$ with $f(0) = e^a$. (For this purpose you could consider $e^x$ as being defined as the solution of $f'(x) = f(x)$ with $f(0) = 1$.) Or $$\sin(x+a) = \sin x\cos a + \cos x\sin a$$ since both sides satisfy $f''(x) = -f(x)$ with $f(0) = \sin a$ and $f'(0) = \cos a$. (For this purpose, $\sin x$ is the solution of $f''(x) = -f(x)$ with $f(0) = 0$ and $f'(0) = 1$, and $\cos x = (\sin x)'$.)
By far the most important application of ODEs is the Fundamental Theorem of Calculus, which reduces the computation of $\int_a^b f(x)\,dx$ to solving the differential equation $y'=f(x)$.