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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.

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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 $\cos x = (\sin x)'$.)

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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 $y'=f(x)$.

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Given a connection on any vector bundle and a path in the base, the Picard-Lindelof theorem tells us that parallel transport is a linear isomorphism between tangent spaces at points along the path. Furthermore, if the path is a loop and the connection is flat, then the isomorphism only depends on the homotopy class of the path. These facts, which all follow from existence and uniqueness of solutions to ODEs, are what make it possible to talk about holonomy groups.

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