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→Existence of solutions
For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point <math>(a,b)</math> in the xy-plane, define some rectangular region <math>Z</math>, such that <math>Z = [l,m]\times[n,p]</math> and <math>(a,b)</math> is in the interior of <math>Z</math>. If we are given a differential equation <math>\frac{dy}{dx} = g(x,y)</math> and the condition that <math>y=b</math> when <math>x=a</math>, then there is locally a solution to this problem if <math>g(x,y)</math> and <math>\frac{\partial g}{\partial x}</math> are both continuous on <math>Z</math>. This solution exists on some interval with its center at <math>a</math>. The solution may not be unique. (See Ordinary differential equation for other results.)
For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point <math>(a,b)</math> in the xy-plane, define some rectangular region <math>Z</math>, such that <math>Z = [l,m]\times[n,p]</math> and <math>(a,b)</math> is in the interior of <math>Z</math>. If we are given a differential equation <math>\frac{dy}{dx} = g(x,y)</math> and the condition that <math>y=b</math> when <math>x=a</math>, then there is locally a solution to this problem if <math>g(x,y)</math> and <math>\frac{\partial g}{\partial x}</math> are both continuous on <math>Z</math>. This solution exists on some interval with its center at <math>a</math>. The solution may not be unique. (See Ordinary differential equation for other results.)
对于一阶初值问题，皮亚诺存在性定理给出了一组解存在的情况。给定的x-y平面上的任意点<math>(a,b)</math>，定义一些矩形区域<math>Z</math>，比如说，<math>Z = [l,m]\times[n,p]</math>而且<math>(a,b)</math>是<math>Z</math>内部一点。如果我们给出一个微分方程<math>\frac{dy}{dx} = g(x,y)</math> 和条件当<math>x=a</math>时<math>y=b</math>，那么如果<math>g(x,y)</math>和<math>\frac{\partial g}{\partial x}</math>在<math>Z</math>上是连续的，那么这个问题就有一个局部解。这个解决方案以<math>a</math>为中心的某些区间上存在。方程的解可能不是唯一的。(其他结果请参见常微分方程。)
对于一阶初值问题，皮亚诺存在性定理给出了一组解存在的情况。给定的x-y平面上的任意点 <math>(a,b)</math> ，定义一些矩形区域 <math>Z</math> ，比如说，<math>Z = [l,m]\times[n,p]</math> 而且 <math>(a,b)</math> 是 <math>Z</math> 内部一点。如果我们给出一个微分方程 <math>\frac{dy}{dx} = g(x,y)</math> 和条件：当<math>x=a</math>时<math>y=b</math>，那么如果<math>g(x,y)</math>和<math>\frac{\partial g}{\partial x}</math>在<math>Z</math>上是连续的，那么这个问题就有一个局部解。这个解决方案以 <math>a</math> 为中心的某些区间上存在。方程的解可能不是唯一的。(其他结果请参见常微分方程。)
For any nonzero <math>f_{n}(x)</math>, if <math>\{f_{0},f_{1},\cdots\}</math> and <math>g</math> are continuous on some interval containing <math>x_{0}</math>, <math>y</math> is unique and exists.
For any nonzero <math>f_{n}(x)</math>, if <math>\{f_{0},f_{1},\cdots\}</math> and <math>g</math> are continuous on some interval containing <math>x_{0}</math>, <math>y</math> is unique and exists.
对于任意非零的<math>f_{n}(x)</math>，如果<math>\{f_{0},f_{1},\cdots\}</math> 和 <math>g</math>在某个包含<math>x_{0}</math>的区间上连续，则<math>y</math>是存在且唯一的。
对于任意非零的 <math>f_{n}(x)</math> ，如果<math>\{f_{0},f_{1},\cdots\}</math> 和 <math>g</math>在某个包含<math>x_{0}</math>的区间上连续，则<math>y</math>是存在且唯一的。
==Related concepts==
==Related concepts==