数模论坛

 找回密码
 注-册-帐-号
搜索
热搜: 活动 交友 discuz
查看: 14839|回复: 3

数学专业英语-First Order Differential Equations

[复制链接]
发表于 2004-5-6 09:30:47 | 显示全部楼层 |阅读模式
< ><FONT face="Times New Roman" size=3>A differential equation is an equation between specified derivatives of a function, its</FONT></P>
< ><FONT face="Times New Roman" size=3>valves,and known quantities.Many laws of physics are most simply and naturally formu-</FONT></P>
< ><FONT face="Times New Roman" size=3>lated as differential equations (or DE’s, as we shall write for short).For this reason,DE’s </FONT></P>
<P ><FONT face="Times New Roman" size=3>have been studies by the greatest mathematicians and mathematical physicists since the </FONT></P>
<P ><FONT face="Times New Roman" size=3>time of Newton..</FONT></P>
<P ><FONT face="Times New Roman" size=3>Ordinary differential equations are DE’s whose unknowns are functions of a single va-</FONT></P>
<P ><FONT face="Times New Roman" size=3>riable;they arise most commonly in the study of dynamic systems and electric networks.</FONT></P>
<P ><FONT face="Times New Roman" size=3>They are much easier to treat than partial differential equations,whose unknown functions</FONT></P>
<P ><FONT face="Times New Roman" size=3>depend on two or more independent variables.</FONT></P>
<P ><FONT face="Times New Roman" size=3>Ordinary DE’s are classified according to their order. The order of a DE is defined as </FONT></P>
<P ><FONT face="Times New Roman" size=3>the largest positive integer, n, for which an n-th derivative occurs in the equation. This</FONT></P>
<P ><FONT face="Times New Roman" size=3>chapter will be restricted to real first order DE’s of the form</FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">               </FONT>Φ<FONT face="Times New Roman">(x, y, y</FONT>′<FONT face="Times New Roman">)=0                                         (1)</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">Given the function </FONT>Φ<FONT face="Times New Roman">of three real variables, the problem is to determine all real functions y=f(x) which satisfy the DE, that is ,all solutions of(1)in the following sense.</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3><B>DEFINITION  </B> A solution of (1)is a differentiable function f(x) such that  </FONT></FONT></P>
<P ><FONT size=3>Φ<FONT face="Times New Roman">(x. f(x),f</FONT>′<FONT face="Times New Roman">(x))=0 for all x in the interval where f(x) is defined.</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3><B>EXAMPLE 1.</B>  In the first-other DE</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">              x+yy</FONT>′<FONT face="Times New Roman">=0                                            (2)</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">the function </FONT>Φ<FONT face="Times New Roman"> is a polynomial function </FONT>Φ<FONT face="Times New Roman">(x, y, z)=x+ yz of three variables in-</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>volved. The solutions of (2) can be found by considering the identity</FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">d(x</FONT>&sup2;<FONT face="Times New Roman">+y</FONT>&sup2;<FONT face="Times New Roman">)/d x=2(x+yy</FONT>ˊ<FONT face="Times New Roman">).From this identity,one sees that x</FONT>&sup2;<FONT face="Times New Roman">+y</FONT>&sup2;<FONT face="Times New Roman"> is a con-</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>stant if y=f(x) is any solution of (2).</FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">The equation x</FONT>&sup2;<FONT face="Times New Roman">+y</FONT>&sup2;<FONT face="Times New Roman">=c defines y implicitly as a two-valued function of x,</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>for any positive constant c.Solving for y,we get two solutions,the(single-valued)</FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">functions y=</FONT>±<FONT face="Times New Roman">(c-x</FONT>&sup2;<FONT face="Times New Roman">)<SUP>0.5</SUP> ,for each positive constant c.The graphs of these so-</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>lutions,the so-called solution curves,form two families of scmicircles,which fill the upper half-plane y&gt;0 and the lower half-plane y&gt;0,respectively.</FONT></P>
<P ><FONT face="Times New Roman" size=3>On the x-axis,where y=0,the DE(2) implies that x=0.Hence the DE has no solutions</FONT></P>
<P ><FONT face="Times New Roman" size=3>which cross the x-axis,except possibly at the origin.This fact is easily overlooked,</FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">because the solution curves appear to cross the x-axis;hence y</FONT>ˊ<FONT face="Times New Roman">does not exist,and the DE (2) is not satisfied there.</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">The preceding difficulty also arises if one tries to solve the DE(2)for y</FONT>ˊ<FONT face="Times New Roman">. Dividing through by y,one gets y</FONT>ˊ<FONT face="Times New Roman">=-x/y,an equation which cannot be satisfied if y=0.The preceding difficulty is thus avoided if one restricts attention to regions where the DE(1) is normal,in the following sense.</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3><B>   DEFINITION. </B> A normal first-order DE is one of the form</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">       y</FONT>ˊ<FONT face="Times New Roman">=F(x,y)                                                  (3)</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">In the normal form y</FONT>ˊ<FONT face="Times New Roman">=-x/y of the DE (2),the function F(x,y) is continuous in the upper half-plane y&gt;0 and in the lower half-plane where y&lt;0;it is undefined on the x-axis.</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><B><FONT face="Times New Roman"><FONT size=3>Fundamental Theorem of the Calculus.<p></p></FONT></FONT></B></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3>The most familiar class of differential equations consists of the first-order DE’s of the form  </FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">                   y</FONT>ˊ<FONT face="Times New Roman">=g(x)                                            (4)</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>Such DE’s are normal and their solutions are descried by the fundamental thorem of the calculus,which reads as follows.</FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3><B>FUNDAMENTAL THEOREM OF THE CALCULUS</B>. Let the function g(x)in DE(4) be continuous in the interval a&lt;x&lt;b.Given a number c,there is one and only one solution f(x) of the DE(4) in the interval such that f(a)=c. This solution is given by the definite integral</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">f(x)=c+</FONT>∫<FONT face="Times New Roman"><SUB>a</SUB><SUP>x</SUP>g(t)dt ,       c=f(a)                                    (5)</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>This basic result serves as a model of rigorous formulation in several respects. First,it specifies the region under consideration,as a vertical strip a&lt;x&lt;b in the xy-plane.Second,it describes in precise terms the class of functions g(x) considered.And third, it asserts the existence and uniqueness of a solution,given the “initial condition”f(a)=c.</FONT></P>
<P ><FONT face="Times New Roman" size=3>We recall that the definite integral </FONT></P>
<P ><FONT size=3>∫<FONT face="Times New Roman"><SUB>a</SUB><SUP>x</SUP>g(t)dt=lim(max</FONT>Δ<FONT face="Times New Roman">t<SUB>k</SUB>-&gt;0)</FONT>Σ<FONT face="Times New Roman">g(t<SUB>k</SUB>)</FONT>Δ<FONT face="Times New Roman">t<SUB>k </SUB>,     </FONT>Δ<FONT face="Times New Roman">t<SUB>k</SUB>=t<SUB>k</SUB>-t<SUB>k-1 </SUB>                 (5</FONT>ˊ<FONT face="Times New Roman">)</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">is defined for each fixed x as a limit of Ricmann sums; it is not necessary to find a formal expression for the indefinite integral </FONT>∫<FONT face="Times New Roman"> g(x) dx to give meaning to the definite integral </FONT>∫<FONT face="Times New Roman"><SUB>a</SUB><SUP>x</SUP>g(t)dt,provided only that g(t) is continuous.Such functions as the error function crf x =(2/(</FONT>π<FONT face="Times New Roman">)<SUP>0.5</SUP>)</FONT>∫<FONT face="Times New Roman"><SUB>0</SUB><SUP>x</SUP>e<SUP>-t</SUP></FONT><SUP>&sup2;<FONT face="Times New Roman">  </FONT></SUP><FONT face="Times New Roman">dt and the sine integral function       SI(x)=</FONT>∫<SUB><FONT face="Times New Roman">x</FONT></SUB><SUP>∞</SUP><FONT face="Times New Roman">[(sin t )/t]dt are indeed commonly defined as definite integrals.</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P  align=center><B><FONT face="Times New Roman">Solutions and Integrals<p></p></FONT></B></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">According to the definition given above a solution of a DE is always a function. For example, the solutions of the DE x+yy</FONT>ˊ<FONT face="Times New Roman">=0 in Example I are the functions y=</FONT>±<FONT face="Times New Roman"> (c-x</FONT>&sup2;<FONT face="Times New Roman">)<SUP>0.5</SUP>,whose graphs are semicircles of arbitrary diameter,centered at the origin.The graph of the solution curves are ,however,more easily described by the equation x</FONT>&sup2;<FONT face="Times New Roman">+y</FONT>&sup2;<FONT face="Times New Roman">=c,describing a family of circles centered at the origin.In what sense can such a family of curves be considered as a solution of the DE ?To answer this question,we require a new notion.</FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3><B>DEFINITION.</B>    An integral of DE(1)is a function of two variables,u(x,y),which assumes a constant value whenever the variable y is replaced by a solution y=f(x) of the DE.</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">In the above example, the function u(x,y)=x</FONT>&sup2;<FONT face="Times New Roman">+y</FONT>&sup2;<FONT face="Times New Roman"> is an integral of the DE     x+yy</FONT>ˊ<FONT face="Times New Roman"> =0,because,upon replacing the variable y by any function </FONT>±<FONT face="Times New Roman">( c-x</FONT>&sup2;<FONT face="Times New Roman">)<SUP>0.5</SUP>,we obtain u(x,y)=c.</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>The second-order DE</FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">           d</FONT>&sup2;<FONT face="Times New Roman">x/dt</FONT>&sup2;<FONT face="Times New Roman">=-x                                          (2</FONT>ˊ<FONT face="Times New Roman">)</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>becomes a first-order DE equivalent to (2) after setting dx/dx=y:</FONT></P>
<P ><FONT face="Times New Roman" size=3>y ( dy/dx )=-x                                          (2)</FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">As we have seen, the curves u(x,y)=x</FONT>&sup2;<FONT face="Times New Roman">+y</FONT>&sup2;<FONT face="Times New Roman">=c are integrals of this DE.When the DE (2</FONT>ˊ<FONT face="Times New Roman">)</FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>is interpreted as equation of motion under Newton’s second law,the integrals </FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">c=x</FONT>&sup2;<FONT face="Times New Roman">+y</FONT>&sup2;<FONT face="Times New Roman"> represent curves of constant energy c.This illustrates an important principle:an integral of a DE representing some kind of motion is a quantity that remains unchanged through the motion.</FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
 楼主| 发表于 2004-5-6 09:31:03 | 显示全部楼层
< 0cm 0cm 0pt; TEXT-ALIGN: center" align=center><B><FONT face="Times New Roman">Vocabulary<p></p></FONT></B></P>< 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P>< 0cm 0cm 0pt"><FONT face="Times New Roman">differential equation       </FONT>微分方程<FONT face="Times New Roman">        error function   </FONT>误差函数<FONT face="Times New Roman">              </FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">ordinary differential equation  </FONT>常微分方程<FONT face="Times New Roman">  sine integral function  </FONT>正弦积分函数</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">order    </FONT>阶<FONT face="Times New Roman">,</FONT>序<FONT face="Times New Roman">                              diameter    </FONT>直径</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">derivative    </FONT>导数<FONT face="Times New Roman">                          curve      </FONT>曲线</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">known quantities </FONT>已知量<FONT face="Times New Roman">                     replace    </FONT>替代</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">unknown   </FONT>未知量<FONT face="Times New Roman">                           substitute    </FONT>代入</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">single variable   </FONT>单变量<FONT face="Times New Roman">                       strip        </FONT>带形</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">dynamic system     </FONT>动力系统<FONT face="Times New Roman">                  exact differential   </FONT>恰当微分</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">electric network    </FONT>电子网络<FONT face="Times New Roman">                   line integral      </FONT>线积分</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">partial differential equation     </FONT>偏微分方程<FONT face="Times New Roman">       path of integral     </FONT>积分路径</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">classify    </FONT>分类<FONT face="Times New Roman">                              endpoints        </FONT>端点</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">polynomial     </FONT>多项式<FONT face="Times New Roman">                        general solution   </FONT>通解</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">several variables        </FONT>多变量<FONT face="Times New Roman">                 parameter      </FONT>参数</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">family      </FONT>族<FONT face="Times New Roman">                               rigorous       </FONT>严格的</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">semicircle    </FONT>半圆<FONT face="Times New Roman">                            existence       </FONT>存在性</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">half-plane    </FONT>半平面<FONT face="Times New Roman">                          initial condition   </FONT>初始条件</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">region    </FONT>区域<FONT face="Times New Roman">                               uniqueness      </FONT>唯一性</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">normal   </FONT>正规<FONT face="Times New Roman">,</FONT>正常<FONT face="Times New Roman">                           Riemann sum  </FONT>犁曼加</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">identity    </FONT>恒等<FONT face="Times New Roman">(</FONT>式<FONT face="Times New Roman">)</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P>
 楼主| 发表于 2004-5-6 09:31:20 | 显示全部楼层
< 0cm 0cm 0pt; TEXT-ALIGN: center" align=center><FONT face="Times New Roman"><B>Notes</B><B><p></p></B></FONT></P>< 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; tab-stops: list 18.0pt; mso-list: l29 level1 lfo19"><FONT face="Times New Roman">1.       The order of a DE is defined as the largest positive integral n,for which an nth derivative occurs in the question.</FONT></P>< 0cm 0cm 0pt"><FONT face="Times New Roman">       </FONT>这是另一种定义句型<FONT face="Times New Roman">,</FONT>请参看附录<FONT face="Times New Roman">IV.</FONT>此外要注意<FONT face="Times New Roman">nth derivative </FONT>之前用<FONT face="Times New Roman">an </FONT>不用<FONT face="Times New Roman">a .</FONT></P><P 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; tab-stops: list 18.0pt; mso-list: l29 level1 lfo19"><FONT face="Times New Roman">2.       This chapter will be restricted to real first order differential equations of the form </FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 32.25pt">Φ<FONT face="Times New Roman">(x,y,y</FONT>ˊ<FONT face="Times New Roman">)=0</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 32.25pt">意思是<FONT face="Times New Roman">;</FONT>文章限于讨论形如<FONT face="Times New Roman">  </FONT>Φ<FONT face="Times New Roman">(x,y,y</FONT>ˊ<FONT face="Times New Roman">)=0</FONT>的实一阶微分方程<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 32.25pt">有时可以用<FONT face="Times New Roman">of the type</FONT>代替<FONT face="Times New Roman"> of the form </FONT>的用法<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 32.25pt"><FONT face="Times New Roman">        The equation can be rewritten in the form  y</FONT>ˊ<FONT face="Times New Roman">=F(x,y).</FONT></P><P 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; tab-stops: list 18.0pt; mso-list: l29 level1 lfo19"><FONT face="Times New Roman">3.       Dividing through by y,one gets y</FONT>ˊ<FONT face="Times New Roman">=-x/y,</FONT>…</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">      </FONT>划线短语意思是<FONT face="Times New Roman">:</FONT>全式除以<FONT face="Times New Roman">y</FONT></P><P 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; tab-stops: list 18.0pt; mso-list: l29 level1 lfo19"><FONT face="Times New Roman">4.       As we have seen, the curves u(x,y)=x</FONT>&sup2;<FONT face="Times New Roman">+y</FONT>&sup2;<FONT face="Times New Roman">=c are integrals of this DE</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">     </FONT>这里<FONT face="Times New Roman">x</FONT>&sup2;<FONT face="Times New Roman">+y</FONT>&sup2;<FONT face="Times New Roman">=c </FONT>因<FONT face="Times New Roman">c</FONT>是参数<FONT face="Times New Roman">,</FONT>故此方程代表一族曲线<FONT face="Times New Roman">,</FONT>由此<FONT face="Times New Roman">”</FONT>曲线<FONT face="Times New Roman">”</FONT>这一词要用复数<FONT face="Times New Roman">curves.</FONT></P><P 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; tab-stops: list 18.0pt; mso-list: l29 level1 lfo19"><FONT face="Times New Roman">5.       Their solutions are described by the fundamental theorem of the calculus,which reads as follows.</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">        </FONT>意思是<FONT face="Times New Roman">:</FONT>它们的解由微积分基本定理所描述<FONT face="Times New Roman">,(</FONT>基本定理<FONT face="Times New Roman">)</FONT>可写出如下<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">        </FONT>句中<FONT face="Times New Roman">reads as follows </FONT>就是<FONT face="Times New Roman">”</FONT>写成<FONT face="Times New Roman">(</FONT>读成<FONT face="Times New Roman">)</FONT>下面的样子<FONT face="Times New Roman">”</FONT>的意思<FONT face="Times New Roman">.</FONT>注意<FONT face="Times New Roman">follows</FONT>一词中的<FONT face="Times New Roman">”s”</FONT>不能省略<FONT face="Times New Roman">.</FONT></P>
 楼主| 发表于 2004-5-6 09:31:32 | 显示全部楼层
< 0cm 0cm 0pt; TEXT-ALIGN: center" align=center><FONT face="Times New Roman"><B>Exercise</B><B><p></p></B></FONT></P>< 0cm 0cm 0pt">Ⅰ<FONT face="Times New Roman">.Translate the following passages into Chinese:</FONT></P>< 0cm 0cm 0pt"><FONT face="Times New Roman">  1.A differential M(x,y) dx +N(x,y) dy ,where M, N are real functions of two variables x and y, is called exact in a domain D when the line integral </FONT>∫<FONT face="Times New Roman"><SUB>c </SUB>M(x,y) dx +N(x,y) dy is the same for all paths of integration c in D, which have the same endpoints.</FONT></P><P 0cm 0cm 0pt; TEXT-INDENT: 37.5pt"><FONT face="Times New Roman">Mdx+Ndy is exact if and only if there exists a continuously differentiable function u(x,y) such that M= u/ x, N=u/ y.</FONT></P><P 0cm 0cm 0pt 18.75pt; TEXT-INDENT: -18.75pt; tab-stops: list 18.75pt; mso-list: l38 level1 lfo20"><FONT face="Times New Roman">2.       For any normal first order DE y</FONT>ˊ<FONT face="Times New Roman">=F(x,y)  and any initial x<SUB>0  </SUB>, the initial valve problem consists of finding the solution or solutions of the DE ,for x&gt;x<SUB>0  </SUB>which assumes a given initial valve f(x<SUB>0</SUB>)=c.</FONT></P><P 0cm 0cm 0pt 18.75pt; TEXT-INDENT: -18.75pt; tab-stops: list 18.75pt; mso-list: l38 level1 lfo20"><FONT face="Times New Roman">3.       To show that the initial valve problem is well-set requires proving theorems of existence (there is a solution), uniqueness (there is only one solution) and continuity (the solution depends continuously on the initial value).</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt">Ⅱ<FONT face="Times New Roman">. Translate the following sentences into English:</FONT></P><P 0cm 0cm 0pt 45.75pt; TEXT-INDENT: -24.75pt; tab-stops: list 45.75pt; mso-list: l38 level2 lfo20"><FONT face="Times New Roman">1)           </FONT>因为<FONT face="Times New Roman">y=</FONT>ч<FONT face="Times New Roman">(x) </FONT>是微分方程<FONT face="Times New Roman">dy/ dx=f(x,y)</FONT>的解<FONT face="Times New Roman">,</FONT>故有</P><P 0cm 0cm 0pt 45.75pt"><FONT face="Times New Roman">d</FONT>ч<FONT face="Times New Roman">(x)/dx=f (x,</FONT>ч<FONT face="Times New Roman">(x))</FONT></P><P 0cm 0cm 0pt 45.75pt; TEXT-INDENT: -24.75pt; tab-stops: list 45.75pt; mso-list: l38 level2 lfo20"><FONT face="Times New Roman">2)           </FONT>两边从<FONT face="Times New Roman">x<SUB>0</SUB></FONT>到<FONT face="Times New Roman">x</FONT>取定积分得</P><P 0cm 0cm 0pt 45.75pt">ч<FONT face="Times New Roman">(x)-</FONT>ч<FONT face="Times New Roman">(x<SUB>0</SUB>)=</FONT>∫<FONT face="Times New Roman"><SUB>x0</SUB><SUP>x </SUP> f(x,</FONT>ч<FONT face="Times New Roman">(x)) dx   x<SUB>0</SUB>&lt;x&lt;x<SUB>0</SUB>+h</FONT></P><P 0cm 0cm 0pt 45.75pt; TEXT-INDENT: -24.75pt; tab-stops: list 45.75pt; mso-list: l38 level2 lfo20"><FONT face="Times New Roman">3)           </FONT>把<FONT face="Times New Roman">y<SUB>0</SUB>=</FONT>ч<FONT face="Times New Roman">(x<SUB>0</SUB>)</FONT>代入上式<FONT face="Times New Roman">, </FONT>即有</P><P 0cm 0cm 0pt 45.75pt">ч<FONT face="Times New Roman">(x)=y<SUB>0</SUB>+</FONT>∫<FONT face="Times New Roman"><SUB>x0</SUB><SUP>x  </SUP>f(x,</FONT>ч<FONT face="Times New Roman">(x)) dx           x<SUB>0</SUB>&lt;x&lt;x<SUB>0</SUB>+h</FONT></P><P 0cm 0cm 0pt 45.75pt; TEXT-INDENT: -24.75pt; tab-stops: list 45.75pt; mso-list: l38 level2 lfo20"><FONT face="Times New Roman">4)           </FONT>因此<FONT face="Times New Roman"> y=</FONT>ч<FONT face="Times New Roman">(x) </FONT>是积分方程</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">                        y=y<SUB>0</SUB>+</FONT>∫<FONT face="Times New Roman"><SUB>x0</SUB><SUP>x</SUP> f (x,y) dx</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">         </FONT>定义于<FONT face="Times New Roman">x<SUB>0</SUB>&lt;x&lt;x<SUB>0</SUB>+h </FONT>的连续解<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">    </FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">   </FONT>Ⅲ<FONT face="Times New Roman">.    Translate the following sentences into English:</FONT></P><P 0cm 0cm 0pt 46.5pt; TEXT-INDENT: -24.75pt; tab-stops: list 46.5pt; mso-list: l31 level1 lfo21"><FONT face="Times New Roman">1)           </FONT>现在讨论型如<FONT face="Times New Roman">  y=f (x,y</FONT>ˊ<FONT face="Times New Roman">) </FONT>的微分方程的解<FONT face="Times New Roman">,</FONT>这里假设函数<FONT face="Times New Roman"> f (x, dy/dx)  </FONT>有连续的偏导数<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt 46.5pt; TEXT-INDENT: -24.75pt; tab-stops: list 46.5pt; mso-list: l31 level1 lfo21"><FONT face="Times New Roman">2)           </FONT>引入参数<FONT face="Times New Roman">dy/dx=p, </FONT>则已给方程变为<FONT face="Times New Roman"> y=f (x,p).</FONT></P><P 0cm 0cm 0pt 46.5pt; TEXT-INDENT: -24.75pt; tab-stops: list 46.5pt; mso-list: l31 level1 lfo21"><FONT face="Times New Roman">3)           </FONT>在<FONT face="Times New Roman"> y=f (x,p)   x  p=dy/dx       p= f/ x+f/ p  dp/dx</FONT></P><P 0cm 0cm 0pt 46.5pt; TEXT-INDENT: -24.75pt; tab-stops: list 46.5pt; mso-list: l31 level1 lfo21"><FONT face="Times New Roman">4)           </FONT>这是一个关于<FONT face="Times New Roman">x</FONT>和<FONT face="Times New Roman">p</FONT>的一阶微分方程<FONT face="Times New Roman">,</FONT>它的解法我们已经知道<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt 46.5pt; TEXT-INDENT: -24.75pt; tab-stops: list 46.5pt; mso-list: l31 level1 lfo21"><FONT face="Times New Roman">5)           </FONT>若<FONT face="Times New Roman">(A)</FONT>的通解的形式为<FONT face="Times New Roman">p=</FONT>ч<FONT face="Times New Roman">(x,c) ,</FONT>则原方程的通解为</P><P 0cm 0cm 0pt; TEXT-INDENT: 116.25pt"><FONT face="Times New Roman">y=f (x,</FONT>ч<FONT face="Times New Roman">(x,c)).</FONT></P><P 0cm 0cm 0pt 46.5pt; TEXT-INDENT: -24.75pt; tab-stops: list 46.5pt; mso-list: l31 level1 lfo21"><FONT face="Times New Roman">6)           </FONT>若<FONT face="Times New Roman">(A) </FONT>有型如<FONT face="Times New Roman">x=</FONT>ψ<FONT face="Times New Roman">(x,c)</FONT>的通解<FONT face="Times New Roman">,</FONT>则原方程有参数形式的通解</P><P 0cm 0cm 0pt 21.75pt"><FONT face="Times New Roman">                  x=</FONT>ψ<FONT face="Times New Roman">(p,c)</FONT></P><P 0cm 0cm 0pt 21.75pt"><FONT face="Times New Roman">                  y=f(</FONT>ψ<FONT face="Times New Roman">(p,c)p)</FONT></P><P 0cm 0cm 0pt 21.75pt"><FONT face="Times New Roman">      </FONT>其中<FONT face="Times New Roman">p</FONT>是参数<FONT face="Times New Roman">,c</FONT>是任意常数<FONT face="Times New Roman">.</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P>
您需要登录后才可以回帖 登录 | 注-册-帐-号

本版积分规则

小黑屋|手机版|Archiver|数学建模网 ( 湘ICP备11011602号 )

GMT+8, 2024-3-29 07:54 , Processed in 0.077354 second(s), 18 queries .

Powered by Discuz! X3.4

Copyright © 2001-2021, Tencent Cloud.

快速回复 返回顶部 返回列表