二阶常系数非齐次线性微分方程解法及例题.ppt

上页下页铃结束返回首页上页下页铃结束返回首页上页下页铃结束返回首页上页下页铃结束返回首页上页下页铃结束返回首页上页下页铃结束返回首页上页下页铃结束返回首页上页下页铃结束返回首页上页下页铃结束返回首页上页下页铃结束返回首页上页下页铃结束返回首页上页下页铃结束返回首页一、f(x)?Pm(x)e?x型二、f(x)=elx[Pl(x)coswx+Pn(x)sinwx]型§12.9二阶常系数非齐次线性微分方程上页下页铃结束返回首页方程y???py??qy?f(x)称为二阶常系数非齐次线性微分方程?其中p、q是常数?二阶常系数非齐次线性微分方程的通解是对应的齐次方程的通解y?Y(x)与非齐次方程本身的一个特解y?y*(x)之和?y?Y(x)?y*(x)?一、f(x)?Pm(x)e?x型y*?Q(x)e?x?下页则得y*???py*??qy*=[Q??(x)?(2??p)Q?(x)?(?2?p??q)Q(x)]e?x??[Q??(x)+2?Q?(x)+?2Q(x)]e?x+p[Q?(x)+?Q(x)]e?x+qQ(x)e?x?[Q(x)e?x]???[Q(x)e?x]??q[Q(x)e?x]提示?01设方程y???py??qy?Pm(x)e?x特解形式为Q??(x)?(2??p)Q?(x)?(?2?p??q)Q(x)?Pm(x)?——(＊)02提示?此时?2?p??q?0?要使(＊)式成立?Q(x)应设为m次多项式?Qm(x)?b0xm?b1xm?1?????bm?1x?bm?(1)如果?不是特征方程r2?pr?q?0的根?则y*?Qm(x)e?x?下页一、f(x)?Pm(x)e?x型y*?Q(x)e?x?设方程y???py??qy?Pm(x)e?x特解形式为Q??(x)?(2??p)Q?(x)?(?2?p??q)Q(x)?Pm(x)?——(＊)则得提示?此时?2?p??q?0?但2??p?0?要使(＊)式成立?Q(x)应设为m?1次多项式?Q(x)?xQm(x)?其中Qm(x)?b0xm?b1xm?1?????bm?1x?bm?(2)如果?是特征方程r2?pr?q?0的单根,则y*?xQm(x)e?x?下页(1)如果?不是特征方程r2?pr?q?0的根?则y*?Qm(x)e?x?一、f(x)?Pm(x)e?x型y*?Q(x)e?x?设方程y???py??qy?Pm(x)e?x特解形式为Q??(x)?(2??p)Q?(x)?(?2?p??q)Q(x)?Pm(x)?——(＊)则得提示:此时?2?p??q?0?2??p?0?要使(＊)式成立?Q(x)应设为m?2次多项式?Q(x)?x2Qm(x)?其中Qm(x)?b0xm?b1xm?1?????bm?1x?bm?(3)如果?是特征方程r2?pr?q?0的重根,则y*?x2Qm(x)e?x?下页(2)如果?是特征方程r2?pr?q?0的单根,则y*?xQm(x)e?x?(1)如果?不是特征方程r2?pr?q?0的根?则y*?Qm(x)e?x?一、f(x)?Pm(x)e?x型y*?Q(x)e?x?设方程y???py??qy?Pm(x)e?x特解形式为Q??(x)?(2??p)Q?(x)?(?2?p??q)Q(x)?Pm(x)?——(＊)则得结论二阶常系数非齐次线性微分方程y???py??qy?Pm(x)e?x有形如y*?xkQm(x)e?x的特解?其中Qm(x)是与Pm(x)同次的多项式?而k按?不是特征方程的根、是特征方程的单根或是特征方程的的重根依次取为0、1或2?下页提示?因为f(x)?Pm(x)e?x?3x?1???0不是特征方程的根?所以非齐次方程的特解应设为y*?b0x?b1?把它代入所给方程?得例1求微分方程y???2y??3y?3x?1的