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Is there a technical reason as to why the existence and uniqueness theorem for ODEs of the form $$y'(x) = F(x,y(x))$$ is proved for initial conditions of the form $$y(x_0) = y_0$$ and not for $$y'(x_0) = y'_0$$

I understand that in most physical applications, only initial values of the form $y(x_0) = y_0$ are present. But is there any other reason for such a form of initial condition, or can we prove existence and uniqueness even for the IVP of the form $y'(x_0) = y'_0$ ?

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A simple reason: if $F(x_0,y)$ is independent of $y$, the initial condition on $y'(x_0)$ gives you no information. Obvious example: $F(x,y)=2x$. Then $$y'(x)=2x,$$ so $y(x)=x^2+C$, and there is no solution unless $y'(x_0)=2x_0$, in which case there are an infinite number of solutions.

(A less trivial example: $F(x,y) = (x-x_0)y$.)

Even if $F(x_0,y)$ is not independent of $y$, you may not be able to determine $y$ uniquely, if $F(x_0,y)$ is not a bijection: consider $F(x,y) = y^2$. Then given $y'(x_0)=a$, both $+a^{1/2}$ and $-a^{1/2}$ are possible values of $y(x_0)$, so you can have 2 solutions, or if $a<0$, possibly no solutions if you want real ones.

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