摘 要
对微分方程中重要的 方程的进行了求解，对2阶常系数非齐次线性微分方程和其他特殊微分方程的进行了求解,省去了求导、代入、化简的过程, 只需简单的代数运算就可求出特解; 由于2阶常系数齐次线性微分方程的通解也是用代数法求解的(只需解特征方程) , 从而得到这类方程的简便解法———代数法. 有些特解是递推公式, 因此, 通过编程用计算机求解这类微分方程将更能体现这种方法的优越性,而直接输出结果. 列出了几种常见特殊形式的2阶非齐次线性微分方程的解法, 用这些方法解方程时仅仅做了1些简单的代数运算, 因此可以称之为代数法,它能大大提高求解这些特殊类型的微分方程的速度和准确性, 同时, 很多实际问题(比如, 电学和力学上的1些问题) 都归结为这类方程的研究,因而,给出它的简便解法, 还具有重要的实际意义. 另外把常微分方程算子解法推广到偏微分方程初值问题,并有简便而实用的求解方法。
关键字； 通解 特解 变量分离 代数法 微分算子Abstract
The important equation has carried on the solution to the differential equation in, often the coefficient nonhomogenous linear differential equation equation and other special differential equations has carried on the solution to two steps,Omitted has asked to lead, the substitution, the simplification process, Only needs the simple algebraic operation to be possible to extract the particular integral; Because two steps often the coefficient uneven sublinear differential equation understands thoroughly also is solves with the algebra law (only must solve secular equation), Thus obtains this class equation the simple solution algebra law Some particular integrals are the recurrence formulas, Therefore, Will solve this kind of differential equation through the programming with the computer to be able to manifest this method the superiority,But direct output result Has listed several kind of common special forms two steps nonhomogenous linear differential equations equations solution, With these method solution equations when has made some simple algebraic operations merely, Therefore may call it the algebra law,It can enhance greatly solves these special types the differential equation speed and the accuracy, At the same time, Very many actual problem (for instance, In the electricity and mechanics some questions) all sum up as this class equation research,Thus,Produces its simple solution, Also has the important practical significance ，Moreover promotes the ordinary differential equation operator solution to the partial differential equation initial-value problem,And easily has but the practical solution method
Key words; Understands thoroughly Particular integral Variable separation
Algebra law Derivative operator
目 录
第1章 Riccati方程的变量分离法求解 4
1．1; 主要结果 4
1．2 运用 7
第2章 特殊形式的微分方程的解法 92．2 2阶常系数非齐次线性微分方程命题 2 11
2．3 2阶常系数非齐次线性微分方程命题 3 12
2．4 2阶常系数非齐次线性微分方程命题 4 13
2．5 2阶常系数非齐次数性微分方程命题 5 15
第3章 算子解法解偏微分方程 16
3．1 微分方程算子解法简介 16
3．2 算子解法求解偏微分方程初值问题探索 17
第4章 毕业设计的总结 19
