文档介绍：§5 全微分方程1使得若存在),,(yxu一、),(),(),(??.0),(),(为全微分方程则称????????全微分方程内有一阶连在单连通区域和当DyxQyxP),(),(,),(),(??dyyxQdxyxP(1) ????yyxxdyyxQxdyxPyxu00),(),(),(0全微分方程,),(),(000xdyxPdyyxQxxyy????;),(Cyxu??0),(?)3()3(2323的通解求方程????dyyxydxxyx解,6xQxyyP???????是全微分方程.?????yxdyyxdxyxyxu03023)3(),(.42344224Cyyxx???原方程的通解为,42344224yyxx???例14解,64xQyxyP???????是全微分方程将左端重新组合)32(14232dyyxdxyxdyy??)()1(32yxdyd???.132Cyxy???原方程的通解为),1(32yxyd???.0324223的通解求方程???dyyxydxyx例2(2))1(222的通解?????dyyxdxyxx解将方程左端重新组合,有例3 求微分方程,02222?????dyyxdxyxxxdx,0)()(2222?????dyyxxdyxxd,0)()(222????yxdyxxd原方程的通解为.)(322322Cyxx???6二、积分因子法定义:问题: 如何求方程的积分因子?7(1)公式法:,)()(xQyP????????xQxQyPyP???????????????,两边同除?xQyPyPxQ?????????????lnln求解不易特殊地:;.有关时只与当xa?,0???y?,dxdx?????8;.有关时只与当yb?)(1lnxQyPQdxd????????)(xf?.)()(???dxxfex?,0???x?,dydy?????)(1lnyPxQPdyd????????)(yg?.)()(???dyygey?:凭观察凑微分得到),(yx?常见的全微分表达式???????????222yxdydyxdx??xydxyydxxdyln????????????)ln(212222yxdyxydyxdx????????xydxydxxdy2?????????xydyxydxxdyarctan22???????????yxyxdyxydxxdyln212210