958/ and 中国斜学我术大学 数理方程复习参考手册 班级：2020春数理方程08班 授课教师：谢如龙老师 课程助教：高源张舒博 2020春数理方程08班内部材料，仅供学习交流使用
数理方程复习参考手册 班 级：2020 春数理方程 08 班 授课教师：谢如龙老师 课程助教：高源 张舒博 2020 春数理方程 08 班内部材料，仅供学习交流使用
数理方程复习参考手册 2020春数理方程08班 目录 1期末考试题目考点分析 4 1.1偏微分方程通解的求解 4 1.1.12014春数理方程B期末第一题 1.1.22017春数理方程B期末第一题. 4 1.2通解法（行波法）求解定解问题. 4 1.2.12016春数理方程B期末第三题. 4 1.2.22017春数理方程B期末第二题. 5 1.2.32019春数理方程B期末第二题. 1.2.42020春数理方程B毕业班期末第一题. 6 1.3求解固有值问题 5 1.3.12014春数理方程B期末第二题. 6 1.3.22016春数理方程B期末第一题 1.3.32016春数理方程B期末第八题. 6 1.3.42019春数理方程B期末第三题. 6 1.3.52020春数理方程B毕业班期末第二题. 6 1.4分离变量法求解定解问题. > 14.12014春数理方程B期末第五题. 14.22014春数理方程B期末第六题. 7 1.4.32014春数理方程B期末第七题 7 1.4.42016春数理方程B期末第二题. 1.4.52016春数理方程B期末第五题. 8 1.4.62016春数理方程B期末第七题. 8 1.4.72017春数理方程B期末第三题. 8 1.4.82017春数理方程B期末第四题. 8 1.4.92017春数理方程B期末第五题 9 1.4.102019春数理方程B期末第四题. 9 1
数理方程复习参考手册 2020 春数理方程 08 班 目录 1 期末考试题目考点分析 ·········································································· 4 1.1 偏微分方程通解的求解 ································································· 4 1.1.1 2014 春数理方程 B 期末第一题 ·············································· 4 1.1.2 2017 春数理方程 B 期末第一题 ·············································· 4 1.2 通解法 (行波法) 求解定解问题 ······················································· 4 1.2.1 2016 春数理方程 B 期末第三题 ·············································· 4 1.2.2 2017 春数理方程 B 期末第二题 ·············································· 5 1.2.3 2019 春数理方程 B 期末第二题 ·············································· 5 1.2.4 2020 春数理方程 B 毕业班期末第一题 ····································· 5 1.3 求解固有值问题 ·········································································· 5 1.3.1 2014 春数理方程 B 期末第二题 ·············································· 6 1.3.2 2016 春数理方程 B 期末第一题 ·············································· 6 1.3.3 2016 春数理方程 B 期末第八题 ·············································· 6 1.3.4 2019 春数理方程 B 期末第三题 ·············································· 6 1.3.5 2020 春数理方程 B 毕业班期末第二题 ····································· 6 1.4 分离变量法求解定解问题 ······························································ 7 1.4.1 2014 春数理方程 B 期末第五题 ·············································· 7 1.4.2 2014 春数理方程 B 期末第六题 ·············································· 7 1.4.3 2014 春数理方程 B 期末第七题 ·············································· 7 1.4.4 2016 春数理方程 B 期末第二题 ·············································· 7 1.4.5 2016 春数理方程 B 期末第五题 ·············································· 8 1.4.6 2016 春数理方程 B 期末第七题 ·············································· 8 1.4.7 2017 春数理方程 B 期末第三题 ·············································· 8 1.4.8 2017 春数理方程 B 期末第四题 ·············································· 8 1.4.9 2017 春数理方程 B 期末第五题 ·············································· 9 1.4.10 2019 春数理方程 B 期末第四题 ············································· 9 1
数理方程复习参考手册 2020春数理方程08班 1.4.112019春数理方程B期末第五题.9 1.4122020春数理方程B毕业班期末第三题 9 1.5特殊函数的积分求解（性质、递推公式的应用） 9 1.5.12019春数理方程B期末第八题.10 1.5.22020春数理方程B毕业班期末第五题第一问.10 1.6积分变换法求解定解问题. 10 1.6.12014春数理方程B期末第四题.10 1.6.22016春数理方程B期末第四题 10 1.6.32019春数理方程B期末第六题. 10 1.6.42020春数理方程B毕业班期末第四题 10 1.7求解格林函数，用基本解方法求解定解问题 11 1.7.12014春数理方程B期末第三题 11 1.7.22016春数理方程B期末第六题 11 1.7.32017春数理方程B期末第六题. 1.7.42017春数理方程B期末第七题.11 1.7.52019春数理方程B期末第七题.11 1.7.62020春数理方程B毕业班期末第六题. 12 2重要考点梳理 13 2.1定解问题的书写 13 2.2行波法求解定解问题（并和积分变换法作比较）. 15 23阻尼振动问题中的行波法使用.17 2.4齐次化原理的应用. 19 2.5分离变量法求解定解问题 44. 21 2.6积分变换法求解定解问题 .27 2.76函数的性质. 28 2.8基本解方法求解定解问题 29 3经典问题专题复习. 30 3.1简介 30 3.2函数变换法的应用 30 3.3重要的物理学定律 32 3.4微元法分析书写定解问题 34 3.5施刘定理相关问题. 36 3.6勒让德多项式的递推公式推导 37 3.7一类重要的函数的傅里叶变换求解的特殊方法. 39
数理方程复习参考手册 2020 春数理方程 08 班 1.4.11 2019 春数理方程 B 期末第五题 ············································· 9 1.4.12 2020 春数理方程 B 毕业班期末第三题 ···································· 9 1.5 特殊函数的积分求解 (性质、递推公式的应用) ··································· 9 1.5.1 2019 春数理方程 B 期末第八题 ·············································· 10 1.5.2 2020 春数理方程 B 毕业班期末第五题第一问 ···························· 10 1.6 积分变换法求解定解问题 ······························································ 10 1.6.1 2014 春数理方程 B 期末第四题 ·············································· 10 1.6.2 2016 春数理方程 B 期末第四题 ·············································· 10 1.6.3 2019 春数理方程 B 期末第六题 ·············································· 10 1.6.4 2020 春数理方程 B 毕业班期末第四题 ····································· 10 1.7 求解格林函数，用基本解方法求解定解问题 ······································ 11 1.7.1 2014 春数理方程 B 期末第三题 ·············································· 11 1.7.2 2016 春数理方程 B 期末第六题 ·············································· 11 1.7.3 2017 春数理方程 B 期末第六题 ·············································· 11 1.7.4 2017 春数理方程 B 期末第七题 ·············································· 11 1.7.5 2019 春数理方程 B 期末第七题 ·············································· 11 1.7.6 2020 春数理方程 B 毕业班期末第六题 ····································· 12 2 重要考点梳理 ······················································································ 13 2.1 定解问题的书写 ·········································································· 13 2.2 行波法求解定解问题 (并和积分变换法作比较) ··································· 15 2.3 阻尼振动问题中的行波法使用 ························································ 17 2.4 齐次化原理的应用 ······································································· 19 2.5 分离变量法求解定解问题 ······························································ 21 2.6 积分变换法求解定解问题 ······························································ 27 2.7 δ 函数的性质 ·············································································· 28 2.8 基本解方法求解定解问题 ······························································ 29 3 经典问题专题复习 ················································································ 30 3.1 简介 ························································································· 30 3.2 函数变换法的应用 ······································································· 30 3.3 重要的物理学定律 ······································································· 32 3.4 微元法分析书写定解问题 ······························································ 34 3.5 施刘定理相关问题 ······································································· 36 3.6 勒让德多项式的递推公式推导 ························································ 37 3.7 一类重要的函数的傅里叶变换求解的特殊方法 ··································· 39 2
数理方程复习参考手册 2020春数理方程08班 3.8分离变量法求解基本解 40 3.9方向导数. 3.10变量代换的定义域问题 44 4课程作业易错题总结
数理方程复习参考手册 2020 春数理方程 08 班 3.8 分离变量法求解基本解 ································································· 40 3.9 方向导数 ··················································································· 42 3.10 变量代换的定义域问题 ································································ 44 4 课程作业易错题总结 ············································································· 46 3
数理方程复习参考手册 2020春数理方程08班 期末考试题目考点分析 1.1偏微分方程通解的求解 偏微分方程通解的求解是重要考点之一，主要通过通解法求解定解问题类型题目考察， 偶尔也会直接出现求解偏微分方程通解的题目.主要掌握三类偏微分方程通解的求解方 法，即掌握可直接积分类型的求解以及变量代换和函数变换方法在转化到可直接积分 类型中的应用.对应作业题目第一章第2题和第6题. 1.1.12014春数理方程B期末第一题 求下列偏微分方程的通解u=u(红，) 1. d0= +=w 1.1.22017春数理方程B期末第一题 求方程wr+yuy=0的一般解. 1.2通解法（行波法）求解定解问题 这类问题几乎每年都有考察，主要考察对于通解法求解定解问题的掌握.这类题目以考 察行波法为主，偶尔会涉及其他类型的偏微分方程通解的求解.需要熟练掌握行波法的 使用条件，以及通解法求解定解问题的流程.对应于作业第一章题目的第11题. 1.2.12016春数理方程B期末第三题 考虑初值问题 4=4u+f,x),(t>0,-<x<+o∞） 4=0=z2,4l=0=sin2z
2020 春数理方程 08 班 数理方程复习参考手册 2020 春数理方程 08 班 期末考试题目考点分析 1.1 偏微分方程通解的求解 偏微分方程通解的求解是重要考点之一，主要通过通解法求解定解问题类型题目考察， 偶尔也会直接出现求解偏微分方程通解的题目. 主要掌握三类偏微分方程通解的求解方 法，即掌握可直接积分类型的求解以及变量代换和函数变换方法在转化到可直接积分 类型中的应用. 对应作业题目第一章第 2 题和第 6 题. 1.1.1 2014 春数理方程 B 期末第一题 求下列偏微分方程的通解 u = u(x, y) 1. ∂ 2u ∂x∂y = x 2 y 2. y ∂ 2u ∂x∂y + ∂u ∂x = xy 1.1.2 2017 春数理方程 B 期末第一题 求方程 ux + yuxy = 0 的一般解. 1.2 通解法 (行波法) 求解定解问题 这类问题几乎每年都有考察，主要考察对于通解法求解定解问题的掌握. 这类题目以考 察行波法为主，偶尔会涉及其他类型的偏微分方程通解的求解. 需要熟练掌握行波法的 使用条件，以及通解法求解定解问题的流程. 对应于作业第一章题目的第 11 题. 1.2.1 2016 春数理方程 B 期末第三题 考虑初值问题 ( utt = 4uxx + f(t, x),(t > 0, −∞ < x < +∞) u| t=0 = x 2 , ut | t=0 = sin 2x 4