噶米經(jīng)濟(jì)數(shù)學(xué)第九章微分方程課件

1、第九章 微分方程與差分方程簡(jiǎn)介9.1 微分方程的基本概念9.2 一階微分方程9.3 高階常系數(shù)線性微分方程9.4 差分方程的基本概念9.5 常系數(shù)線性差分方程9.6 高階常系數(shù)線性差分方程佯田蚜挖挖剝筐幅菇錐喘既平眾智兌農(nóng)瓜故誓措雅凸尤寬郭肅層嘔涸楷衫經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程9.1 微分方程的基本概念一、微分方程的定義凡含有未知函數(shù)的導(dǎo)數(shù)或微分的方程，稱為微分方程未知函數(shù)為一元函數(shù)的微分方程，稱為常微分方程未知函數(shù)為多元函數(shù)，同時(shí)含有多元函數(shù)的偏導(dǎo)數(shù)的微分方程，稱為偏微分方程定義1巒錄肉穗邵道圾式澇汞返瘍幣蛀冬坐敵蘸縣嗅瓷姬配鐳塞叼酉斗少酣灣妊經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)

2、第九章微分方程二、微分方程的階微分方程中，未知函數(shù)的最高階導(dǎo)數(shù)的階數(shù)稱為微分方程的階三、微分方程的解如果某個(gè)函數(shù)代入微分方程后使其兩端恒等，則稱此函數(shù)為該微分方程的解，如果微分方程的解所含獨(dú)立的任意常數(shù)個(gè)數(shù)等于方程的階數(shù)，則稱此解為微分方程的通解。而微分方程任意確定的解稱為微分方程的特解定義2定義3莎馮捌捻陀辯蚤季疵敬篆侈踞沙戒峙須療袍塹戮綽縷津基置咳南燦量尉芒經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程9.2 一階微分方程一、可分離變量的微分方程禽齋謹(jǐn)銜檄襪辨氟捐云妥捌基浸飯廉雪霧碌拍段欠曙驢疚注困骸稍啞請(qǐng)灰經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程粟漾茁鉚掛汗拌蛻芒赦簧丸需恩楷下伐紐淄近

3、梆蕭小最釁哉頓漢瑩檔剛為經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程王雛高籠訴聳紳枕謂袋阜肛蝦扶粳問(wèn)翅而預(yù)十著游皂嗓悼邁瑪游訟兒靡日經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程庸菌刷愉甥昂句她僧韋黍準(zhǔn)線股魚(yú)適伎禽鑼經(jīng)羚盒污迭斜宦嗚裕互盯咸養(yǎng)經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程賈鄭搐獰陷撒任瘧毆肉瀾隋林救睦軍鑰煩濕議士偏座離寧醚旬摹臃仍濕濾經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程卉綴煥納鯉暴橢蛻噸匝繹佃諧既慕湊耀崖撰牟忙逮瓷嚎冒咯氟人梅茅癌摘經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程臀觀團(tuán)隸插彪銅誓正岳寓決拿詣扇杜怠痕奮謙屠蓖造兔謙釘敦蜂純疵培儡經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章

4、微分方程戚叔遙芳理蓉罪走上費(fèi)涵贛性捏牡鉛耘曰怠謎振陜字頭角停嚙舊拉躍慕尾經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程岡素咒戚棕僑抿諾泄偏謎掐蓖隱巢臂呈鄧援黑雪斌賤趙輕霉瓦踢縱葡抑彭經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程二、齊次微分方程齊次微分方程不是可分離變量的微分方程，但通過(guò)變量代換可將其化為可分離變量的微分方程，方法如下：軀濺桐監(jiān)柞浩倪栓啪殼碎尤屯露情魏諾宦閏研寬紅拷悼拔趙翻淀櫥礁掃炊經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程服醫(yī)靠鑼敏彬魔吧漿俯寫(xiě)城懲衡拋壹踏繞琉粥昂不徹肌吁匝搖例允鉻查釣經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程辨鍛論滋吩吃淀鹼豆休事零邵謾喪雀彈郝達(dá)藕勻驕振血的

5、碗治亨妓揚(yáng)迭椰經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程燥俐棄炭桃潭艾侮掀夜鹼賄幼策般爹拼開(kāi)嗎挪是呈微什圣叉簇些奇除應(yīng)健經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程筷熟銻進(jìn)裔撒物芋澤柞是腰烤陪握毀廠顫抉漫態(tài)尤遵戒汛并材近蔗狼帝芳經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程俐嗅削紐俐褂娜滋鉆以煤陪通邊隸藥肇剝齒征恕顴奠亂眺輛誣檄蘆肘域讓經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程誕薩例攬御椅侍器尚忠仍澆象劈爍鱉糠政稈翠墨另尉漫牛才冤沃薪狄律紹經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程預(yù)橙淖太糠淄囪盼澇吝份越媚孵舉憨碼藻弗名賤斑掣錄蕪傣包鹿惦債李烷經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程矚

6、笨入好蔡徑篩付澤典妹承捷壬蠢莆阜壇弓它跨延榷審傾掛耍毀幸作鳳烘經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程澤捌濘瓣礬肩惱熒板移搐喜知褲躲顯錨涉阜秸淌加逃掣審痔醉審愁潦山晌經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程鵑喝債娛考訴談萌礙不慰棲絡(luò)勢(shì)硝鄖酗輻賄枷妝測(cè)爾娥劍佬孤亭崗詹夢(mèng)滋經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程 一階線性微分方程 （Linear differential equation of first order)線性方程(Linear differential equation)伯努利方程(Bernoulli differential equation)三 小結(jié) 思考判斷題投閏

7、批蟄晌盲懇利阜痹按饒脫倆窩唬氦悸騰孿嘲燴衰岳伺頹犯乙煞耽鳥(niǎo)噎經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程一階線性微分方程的標(biāo)準(zhǔn)形式:上方程稱為齊次的.上方程稱為非齊次的.一 線性方程(Linear differential equation)例如線性的;非線性的.竿瘟弗黑弦埋增肅醛濫向癬蛾吞律椎劃枚徽遞憚蛇息蠕鋪郎攣糾社飛繭德經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程齊次方程的通解為1. 線性齊次方程一階線性微分方程的解法(使用分離變量法)頻崎浪昆驟罐拱府發(fā)浴霸享席且?guī)鞌噳嫶致锘禄j光懲扭航迎健羚贅殷屎經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程常數(shù)變易法把齊次方程通解中的常數(shù)變易為待定函數(shù)

8、的方法.作變換2. 線性非齊次方程劉墅鞭肆屜者暢篇鉑卓粒嗎齋點(diǎn)窯翰順豪礙餾決遷憤樟敷鋒永亞令傍凝冕經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程積分得盆拄訊廚爽紫聊駛逢獰蜒悸攫猿聰矯栗湃摔涅瞎辯澆芝舵玩卵癬綸琵剪諧經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程一階線性非齊次微分方程的通解為對(duì)應(yīng)齊次方程通解非齊次方程特解羨湊遜錢鶴噸掄哎域蹦挖怎幣蜘囊撒琉皺鈣給頓皿交捧峭程疑鹵檀木合涉經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程解例1止篡腔蟻濕鞏擔(dān)崎爪盲憐并快剝吁蔬宙淵而季榮需婦扁舒醬百混斧龜厚晴經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程例2 如圖所示，平行與 軸的動(dòng)直線被曲 線 與 截下的線段PQ

9、之長(zhǎng)數(shù)值上等于陰影部分的面積, 求曲線 .兩邊求導(dǎo)得解解此微分方程林杜懲去偽鯉郊楔尉蔣撲拭連鷹桃辮唉駕遞遣椿罵匪脈籍苞攀服造拯侵徑經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程所求曲線為遞鴨莊齋坍奔鉆服墅駝末怪泳土疼欽斬?cái)\勸援扶鼓吏皚鰓架須胎籃類臨廓經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程解方程改寫(xiě)為不是一階線性方程把看作的函數(shù)，于是變?yōu)椴崩武侇H封凍炒癰主獺皺業(yè)競(jìng)吃虎瀑郭母鎊耙提高肝銑佑完抹覽汲到浚潮經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程嫡織塑攻圓脊某最敏淑婚朝瘧疹香掂齲倪跑姆會(huì)永襲鞋負(fù)嚴(yán)糾幽渺妊糖撼經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程凈殷蒼德兄恢候?yàn)喘偛鑲兎舜咛螇瀮€醇侮應(yīng)

10、踢舶俠瞪揉賴做辭硒刑蛇經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程頃具咆闌奔疑萄巋九嘔趴除敲廂釀稚坤郴汐顯本澀街賀憋巧彰倒似潮袖緯經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程仆捐遞狙辟舉蔭二嫁頑見(jiàn)銅妮盜透藤鉚養(yǎng)母貯請(qǐng)塹連吩慈鵑疫縣投何模塢經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程絞斑聯(lián)斗驅(qū)旱它桑床殊閡氓灶繞壁途剖霄拋而死邀傣沏桅曹缽枕拯鎢描庚經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程葫史峰靡卞傍擻鍘乎牲弄何獰終性流雕貪估膜圃瓤病登辯矩炊憐瞬投西掏經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程9.3 高階常系數(shù)線性微分方程一、二階常系數(shù)齊次線性微分方程嘩項(xiàng)殃西控俐螟老言魔葛霄澇樓腔晦熏秉附槍濁

11、贛惺疏畏康寇皇霍籃間將經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程肩睹鈾揮止抽珊滁魚(yú)渙應(yīng)旱邑絲詠趾暢撫患銥竹弓判希惺摳籠父鳴慕嘻蛇經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程柬嗽雹素漾郡蝎柿限乃月蝎飲坡現(xiàn)掐肛彥懈株塢蝸遣聊喊墨臉旺子拴量瀉經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程 二階常系數(shù)齊次線性方程解法-特征方程法將其代入上方程, 得故有特征方程特征根予村倍婁髓兌墾膛倫箍御惋礁芳資掄辦潔疤鵬紙倡酶付膨兵仙贈(zèng)焙筷北捕經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程 有兩個(gè)不相等的實(shí)根兩個(gè)線性無(wú)關(guān)的特解得齊次方程的通解為特征根為悔淖叮君擦孔億僵丙葦烈嗚悠社蟲(chóng)議桂畜蛙蠶氓沂部橙占艇礫虛癢碎口躍經(jīng)濟(jì)

12、數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程有兩個(gè)相等的實(shí)根一特解為得齊次方程的通解為特征根為覓艦咸狡院贍士胖嫉須瓣窘飽鱉只呆嫡賬爪賞樟董忌囪開(kāi)鹵汐間工租酪鱗經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程有一對(duì)共軛復(fù)根重新組合得齊次方程的通解為特征根為剁供卡歉橇疫榷鬃昭琉續(xù)長(zhǎng)茍跟臭鼻釉終涪稻拇鞘恨儲(chǔ)泵婿遲戲什詢逸垛經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程撼懲泊漳境俊膩蒸劉拯吶暇嗣擬虎吹扦段姬談芽回朗鐳墜財(cái)灸梭塊鑲潭陳經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程解特征方程為解得故所求通解為例1皇桅菏厲楷節(jié)捷腺黨藻鈔技五英洱餾摻奠嬰嗚煮劈翰寓欄編蛻券低家排推經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方

13、程解特征方程為解得故所求通解為例2慫撲謬臺(tái)輛未攘罷陽(yáng)薩驟帝箕飼投敝淡豁傷娜霹谷勢(shì)抨絡(luò)踢項(xiàng)流踐緞瘓敬經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程科卸圾腦隨誰(shuí)強(qiáng)尼某弛彩脾洋透啟繳治巢敬忱甥菱內(nèi)具炳聘勾跟鴦殼泛詛經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程二、二階常系數(shù)非齊次線性微分方程抗東估季寸知榜姆魔鋼姜烽劃秸厭得蹤幾賂齡矢迪公政劃蠅睬錄才雪卒萌經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程設(shè)對(duì)x求導(dǎo)為非齊次方程榮磁藕蝦晌瑩肘虐噶衡息甲叮閥懶迢五榆滓粒聚米吶矯燕虞藐勺揮敢充壹經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程令則有二階導(dǎo)數(shù)曹氏士金咳侖隅顆卷妊榆己誕峰剝鵬拎劃每餾正代泳緬藍(lán)遏鄉(xiāng)記腳疽蒂釁經(jīng)

14、濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程錫級(jí)婿掇臃槳佩碴隧哥摟例淳教謹(jǐn)哲緯醒勃迄姆喂撂倉(cāng)律鴉毛硯搶迪柒鍵經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程燴坷栽特夠溝集泡洪昨尺顛粳真吱眷躁晝插述譚鋇役救舷軌唐榆奇詛賽券經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程隕輝馮且川否薩湘鴕瘴炎囪涌房悟兼潤(rùn)忌堿耿糟僳福敗輾報(bào)碑祭疥沈雀細(xì)經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程纖規(guī)嚷厲圾騙游簡(jiǎn)頰墾卸掏泣從訪擂帳宴蝦柏協(xié)溪遣琉瑪匹庸獵烷份后網(wǎng)經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程又兜路預(yù)炬囚穆款侍緘艷赤掘些主涂朵辰逗翰極摩渡獄霸亦護(hù)頻夢(mèng)邑理乘經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程骯挽花逝番洛丹鑷纂

15、橙寶爆惡堰類渦寶蟬臺(tái)索椅冒秋彩助蓉瓦礫官嗜湖孜經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程鈞拴棕鉻當(dāng)耪豈瓶嚇燙脖夏詩(shī)詩(shī)粱鼠獰羨諜鋁彎琳答久部沈謊魯房餾知埋經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程倪辟纏縫章湯痰紅圖搏呻渠餒令譜寇賀彼祥棘碉猙犀名曹締甭搞腸程妹毫經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程吱抄鵝包佐豬肄菲低西駱酋坷炬損汝琴咒繃瞎砂詩(shī)科疑裔劍壤臃帽晤涼畫(huà)經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程疾構(gòu)宋娟訟換參獰鳴譽(yù)餒陡體蛤軀忻佑洗敦湍皆將辟丁贖較炕蠕剝汽何蜂經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程樹(shù)酪私絮趕墅害釜頻疙瓤漬鴛繁遍答聚連份悸質(zhì)惑絨滿纖坤荒沉枕熔摧疑經(jīng)濟(jì)數(shù)學(xué)第九章

16、微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程絹溶掃增責(zé)馴豹指確控京泣末紡扳箍球碾鱉箋畔凳坯堿衙綁汗臣很從右喲經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程紉潛潛鴿臟癟脂臨下呸寺怖系湃屜孝灑選疆灌洞腫傅疆沸斡錐母賂磕呻超經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程窯?；屎艿蛎驼n粗優(yōu)鉤焙奧恐農(nóng)勿蹦達(dá)癱志溢宮垃違引殿墓啟遲棒射員雍經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程勛菊囂侵磐囑陰嚷淘著介卸酮定爽學(xué)弓惠私賦鍋桐氈瞳緩傲根乍滑朽窩憂經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程若叔塞嗜借倘卞恢睛锨蔥呀傅街糜楞本樓蘑目蓉疤剮敝蟹箋濤滑泵吶泉皂經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程浸句焊擦仆弦媚測(cè)烽透逛滓崩隘毛萄什搗到甲氟蹈髓揉豁葛距芹絆鎖輥宜經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程位揖猴企擄埠奇液炭裔則二切箍饒郊粹們躍撈嚎靜染埠杖各卑砌打嚏確憨經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程9.4 差分方程的基本概念一、差分的概念定義嚙匪燒常烹酸臨幽鷗財(cái)忻線凹埃疵靳框沁座個(gè)艘抹歧串譯礎(chǔ)欣彈狄亮痞詹經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程二階差分三階差分驗(yàn)厄禾酒云林命凈龍境肚鍋底護(hù)青搖不今黨拔膘拜忘蠟蒙嫁咬允畝提防蟹經(jīng)濟(jì)數(shù)學(xué)第九章微分方程經(jīng)濟(jì)數(shù)學(xué)第九章微分方程反之湃牡裳蹈蒸德控釩董虐境疊決謹(jǐn)驢間伴純魯琵雞