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第320行：
第320行： − 解微分方程不同于解代数方程。不仅它们的解往往不清楚，而且解是否唯一或是否存在也是值得关注的课题。+ 第328行：
第328行： − + 第359行：
第359行： − 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。+
→Existence of solutions
Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.
Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.
解微分方程不同于解代数方程。方程解的情况往往是不确定的，而且解是否唯一或是否存在也是值得关注的问题。
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> 和当<math>x=a</math>时<math>y=b</math>，如果<math>g(x,y)</math>和<math>\frac{\partial g}{\partial x}</math>在<math>Z</math>上是连续的，这个问题就有一个局部解。这个解在以 <math>a</math> 为中心的某些区间上存在，其可能不是唯一的。(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> ，定义矩形区域 <math>Z</math> ，如，<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> 和当<math>x=a</math>时<math>y=b</math>，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。这个解在以 <math>a</math> 为中心的某些区间上存在，其可能不是唯一的。(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 <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==