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1、5.1变系数二阶线性齐次常 微分方程的特殊解法 一.引言 二.线性常微分方程的变换性质 三.常系数化法 四.降阶法 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.一.引言 1. 求解二阶
2、线性常微分方程的重要性 2. 困难 3. 解决问题的途径 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.二. 线性常微分方程的变换性 质 设最一般的二阶变系数线性齐次常微分方 程为（5。
3、11） 1. 方程（5。11）对自变量的任意变换 的保线性性 2. 方程（5。11）对未知函数的线性变 换的保线性性 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.三三. .常系数化法常
4、系数化法1.1.通过通过自变量的变换自变量的变换使方程的系数化使方程的系数化 为常数为常数 2.2.通过通过未知函数的齐次线性变换未知函数的齐次线性变换使方使方 程的系数化为常数程的系数化为常数 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyrig
5、ht 2004-2011 Aspose Pty Ltd.例. 将uler型方 程解: 将方程化为标准型(.)(a,b为常数) 常系数化Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.例.2
6、. 将 阶essel方程 ( 为常数)常系数化. 解 :根据判别式若可以经未知函数的线性变换常系数化,只要在 (5.1-12)中取Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.Evalu
7、ation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 1.dAlembert 降阶法设已知一个特解（用观察法）y1,用变换 y=uy1 可以把原方程化为关于u的一阶线 性方程。 2.利用算子因式分解降阶
8、四. 降阶法 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.ENDEvaluation only.Evaluation only. Created with Aspose.Slides
9、for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.1.求解二阶线性常微分方程 的 重要性 这些方程是物理学与科学技术最常见的，有直接应 用；是解高阶线性常微分方程的基础；是解数学物理方程和学习后继课程的基础 。 Evaluation only.Evaluation only. Created with Asp
10、ose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.2. 困难 最一般的二阶变系数线性常微分方程 非常难解，至今没有一般的方法。Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client P
11、rofile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.3. 解决问题的途径 一阶线性常微分方程总是可解的;降阶法化二阶为一阶. 二阶常系数线性常微分方程总是可解的;常系数化法化变系数为常系数.如：著名的Euler方程及其它一些方程。 但是，都没有解决哪些方程可以常系数 化，用什么变换，怎么找到这个变换，变 换成什么样的常系数方程，以便迅速求解 。Eval
12、uation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.1. 1.方程（方程（5 5。1 11 1）对自）对自 变变 量的任意变换的保线性性量的任意变换的保线性性 方程（5。11）化为Evaluation
13、 only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.2. 方程（5。11）对未知 函数的线性变换的保线性性 若=0，上式化为Evaluation only.Evaluation only. Created wit
14、h Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.1.1.通过自变量的变换使方程通过自变量的变换使方程 的系的系数化为常数数化为常数 如Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.
15、5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.2.2.通过未知函数的齐次线性变通过未知函数的齐次线性变 换使换使方程的系数化为常数方程的系数化为常数Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.
16、2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.
