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第312行：
第312行： − + 第319行：
第319行： − 解微分方程不同于解代数方程。不仅他们的解决方案往往不清楚，而且解决方案是否独一无二或是否存在也是值得关注的问题。+ 第327行：
第327行： − 对于一阶初值问题，皮亚诺存在性定理给出了一组存在解的情况。给定 xy 平面上的任意点数学(a，b) / math，定义一些矩形区域数学 z / math，比如说，math z [ l，m ] times [ n，p ] / math (a，b) / math 是在 math z / math 的内部。如果我们给出一个微分方程数学问题 g (x，y) / math 和数学问题 y b / math 当数学问题 x a / math 时的条件，那么如果数学问题 g (x，y) / math 和 frac 部分数学问题都是数学问题 z / math 上的连续问题，那么这个问题就有一个局部解。这个解决方案以数学 a / math 为中心，以某个时间间隔存在。解决方案可能不是唯一的。(其他结果请参见常微分方程。)+ 第335行：
第335行： − 然而，这只能帮助我们解决一阶初始值问题。假设我们有一个 n 阶线性初值问题:+ − − <math> − − 数学 − − f_{n}(x)\frac{d^n y}{dx^n} + \cdots + f_{1}(x)\frac{d y}{dx} + f_{0}(x)y = g(x) − − − F { n }(x) frac { d ^ n y }{ dx ^ n } + cdots + f {1}(x) frac { d y } + f {0}(x) y g (x) − − </math> − − − 数学 第361行：
第347行： − 这样+ − − <math> − − 数学 − − − y(x_{0})=y_{0}, y'(x_{0}) = y'_{0}, y(x_{0}) = y_{0}, \cdots − − Y (x {0}) y {0} ，y’(x {0}) y’{0} ，y (x {0}) y {0} ， cdots − − − </math> − − 数学 − 第387行：
第358行： − 对于任意非零的数学 f { n }(x) / math，如果数学 f {0}、 f {1}、 cdots / math 和数学 g / math 在某个区间上连续，且包含数学 x {0} / math，则数学 y / math 是唯一存在的。+ − −
→Existence of solutions
==Existence of solutions==
==Existence of solutions==
解的存在性
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。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。
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。)
However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:
However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:
然而，这只能帮助我们解决一阶初始值问题。假设我们有一个n阶线性初始值问题:
:<math>
:<math>
f_{n}(x)\frac{d^n y}{dx^n} + \cdots + f_{1}(x)\frac{d y}{dx} + f_{0}(x)y = g(x)
f_{n}(x)\frac{d^n y}{dx^n} + \cdots + f_{1}(x)\frac{d y}{dx} + f_{0}(x)y = g(x)
</math>
</math>
such that
such that
such that
such that
其中有
:<math>
:<math>
y(x_{0})=y_{0}, y'(x_{0}) = y'_{0}, y''(x_{0}) = y''_{0}, \cdots
y(x_{0})=y_{0}, y'(x_{0}) = y'_{0}, y''(x_{0}) = y''_{0}, \cdots
</math>
</math>
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。
==Related concepts==
==Related concepts==