幾類二階變系數(shù)常微分方程解法論文

1、二階變系數(shù)常微分方程幾種解法的探討二階變系數(shù)常微分方程幾種解法的探討胡博（111114109）（湖北工程學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院湖北孝感432000）摘要：摘要：常系數(shù)微分方程是我們目前可以完全解決的一類方程，而求變系數(shù)常微分方程的通解是比較困難的，一般的變系數(shù)常微分方程目前是還沒有通用解法的。本文主要對二階變系數(shù)常微分方程求解進(jìn)行了探究，利用特解、常數(shù)變易法、變量變換等方法求出了某些二階變系數(shù)線性微分方程的通解，并初步歸納了二階變系數(shù)線性方

2、程的求解基本方法及步驟。關(guān)鍵詞關(guān)鍵詞：二階變系數(shù)線性微分方程；變換；通解；特解ToexplethesolutionofsomedinarydifferentialequationsoftwodervariablecoefficientZhangjun(111114128)(SchoolofMathematicsStatisticsHubeiEngineeringUniversityHubeiXiaogan432000)Abstract:

3、Differentialequationwithconstantcoefficientsisaclassofequationswecancompletelysolvethepresentgeneralsolutionchangecoefficientdifferentialequationsisdifficultthevariablecoefficientdinarydifferentialequationisatpresentther

4、eisnogeneralsolution.Thispapermainlyexplesthedinarydifferentialequationwithvariablecoefficientsofdertwotheuseofspecialsolutionsvariationofconstantsvariabletransfmmethodtoextractsometwoderlineardifferentialequationwithvar

5、iablecoefficientsofthegeneralsolutionsummarizesthetwobasicmethodsfsolvingthesecondderlinearequationswithvariablecoefficientssteps.Keywds:Twodervariablecoefficientlineardifferentialequationstransfmationgeneralsolutionspec

6、ialsolutionf()()()erx=0(1.1.3)此時若對存在常數(shù)r使得對一切x恒成立，則方程f()，()，()(1.1.3)有一特解此時要想求出方程的通解，還需要找出另一個特解(1.1)y1=erx(1.1)，且是線性無關(guān)的。y2y1，y2聯(lián)想到常數(shù)變易法，易想到假設(shè)是方程的一特解，y2=u(x)erx也(1.1)則y2=[u(x)ru(x)]erxy2=[u(x)2ru(x)r2u(x)]erx將代入方程得：y2，y2，y

7、2(1.1)f(x)u(x)[2rf(x)p(x)]u(x)[f(x)r2p(x)rq(x)]u(x)=0由于f(x)r2p(x)rq(x)=0?f(x)u(x)[2rf(x)p(x)]u(x)=0(1.1.4)令h則將方程降為一階線性(x)=u(x)h(x)=u(x)(1.5)?1h(x)dh(x)=[?2r?p(x)f(x)]dx?h(x)=e?2rx?∫p(x)f(x)dx即得出dudx=h(x)=e?2rx?∫p(x)f(x)d