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Mathematics
Differential Equations
Existence of unique solutions to a first order ODE on this interval
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[QUOTE="wrobel, post: 6583321, member: 593228"] The following theorem is also useful. Assume that $$f(t,x)\in C^1((t_1,t_2)\times D,\mathbb{R}^m)$$ where ##D\subset\mathbb{R}^m## is an open domain. Assume also that $$|f(t,x)|\le c$$ for all ##(t,x)\in (t_1,t_2)\times D##. Theorem. Let ##x(t)## be a solution to the following IVP $$\dot x=f(t,x),\quad x(t_0)=x_0\in D,\quad t_0\in(t_1,t_2).$$ Assume that ##x(t)## is defined in ##[t_0,t^*),\quad t^*<t_2 ## and can not be extended longer than ##t^*##. Then the following limit exists $$\lim_{t\to t^*-}x(t)=x^*$$ and $$x^*\notin D.$$ [/QUOTE]
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Forums
Mathematics
Differential Equations
Existence of unique solutions to a first order ODE on this interval
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