NTU-Exam 板


LINE

課程名稱︰工程數學二 課程性質︰系必修 課程教師︰王勝仕 開課學院:工學院 開課系所︰化工系 考試日期(年月日)︰2011年4月29日 考試時限(分鐘):140分鐘 是否需發放獎勵金:是 (如未明確表示,則不予發放) 試題 : 1.The ODE is given as follows: x^2y"-4xy'+(x^2+6y)=0 (a)Which type of the power series solution, Taylor series solution or Frobenius series solution, exists at x=0 for the above ODE? Explain why? (b)What are the corresponding indicial equation and recurrence formula/ relation? (c)Find the series solution of the above ODE. 2.Consider a set of linearly independent (but not orthogonal) functions: {v1,v2,v3}, x=[-1,1]. (a)How can you transform the above linearly independent functions into an orthogonal function set {P1,P2,P3}, x=[-1,1]? Show the formula you would use in the transformation process (v1,v2,v3與P1,P2,P3間的關係?) (b)If {v1,v2,v3}={1,x,x^2}, x=[-1,1], please show that these three functions are linearly independent. (c)If {v1,v2,v3}={1,x,x^2}, x=[-1,1], what are the norm of v2? What are P2 and P3? (d)Given that f(x)=e^x=c1P1+c2P2+c3P3, x=[-1,1], find the values of the constants c1,c2 and c3. 3.Considering the following questions (a)Find the general solution of the question: x^2y"-5xy'+9(x^6-8)y=0 (b)Plot the graphs of Y0(x), J0(x) and J1(x) and also show the values of J0(0), J1(0), and Y0(0). (c)Write done the Bessel basic formula, and use Bessel basic formula to obtain Bessel Recurrence Formula and Bessel differential formula. (d)Use J0(x)=A and/or J1(x)=B to express J4(x)and J'0(x). 4.Considering the following questions: (a)Is {sin(nx)} a complete orthogonal set? What would be the corresponding Sturm-Liouville BVP system of the above function set? (b)Why do we have to learn Sturm-Liouville BVP? How many non-trivial solution (or eigenfunctions) can we obtain from Sturm-Liouville BVP? What are the key properties for the eigenfunctions and eigenvalues obtained from Sturm-Liouville BVP? What purposes/functions do these eugenfunctions serve? (c)In general, we can expand a function using the generalized Fourier series and Tylor series. What are the differences between these two series expansions? (d)Can the equation (*) be transformed into Sturm-Liouvile type equation? How? What is the type of the boundary condition (**)? Find the eigenvalues and corresponding eigenfunctions of the above Sturm-Liouville BVP system (*) + (**). y"-2y'+(λ+1)y=0 (*) y(0)=y(4)=0 (**) Please also show that any two different eigenfunctions (with different eigenvalues) are orthogonal. --



※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.245.85 ※ 編輯: mgla8063 來自: 140.112.245.85 (06/25 00:39)







like.gif 您可能會有興趣的文章
icon.png[問題/行為] 貓晚上進房間會不會有憋尿問題
icon.pngRe: [閒聊] 選了錯誤的女孩成為魔法少女 XDDDDDDDDDD
icon.png[正妹] 瑞典 一張
icon.png[心得] EMS高領長版毛衣.墨小樓MC1002
icon.png[分享] 丹龍隔熱紙GE55+33+22
icon.png[問題] 清洗洗衣機
icon.png[尋物] 窗台下的空間
icon.png[閒聊] 双極の女神1 木魔爵
icon.png[售車] 新竹 1997 march 1297cc 白色 四門
icon.png[討論] 能從照片感受到攝影者心情嗎
icon.png[狂賀] 賀賀賀賀 賀!島村卯月!總選舉NO.1
icon.png[難過] 羨慕白皮膚的女生
icon.png閱讀文章
icon.png[黑特]
icon.png[問題] SBK S1安裝於安全帽位置
icon.png[分享] 舊woo100絕版開箱!!
icon.pngRe: [無言] 關於小包衛生紙
icon.png[開箱] E5-2683V3 RX480Strix 快睿C1 簡單測試
icon.png[心得] 蒼の海賊龍 地獄 執行者16PT
icon.png[售車] 1999年Virage iO 1.8EXi
icon.png[心得] 挑戰33 LV10 獅子座pt solo
icon.png[閒聊] 手把手教你不被桶之新手主購教學
icon.png[分享] Civic Type R 量產版官方照無預警流出
icon.png[售車] Golf 4 2.0 銀色 自排
icon.png[出售] Graco提籃汽座(有底座)2000元誠可議
icon.png[問題] 請問補牙材質掉了還能再補嗎?(台中半年內
icon.png[問題] 44th 單曲 生寫竟然都給重複的啊啊!
icon.png[心得] 華南紅卡/icash 核卡
icon.png[問題] 拔牙矯正這樣正常嗎
icon.png[贈送] 老莫高業 初業 102年版
icon.png[情報] 三大行動支付 本季掀戰火
icon.png[寶寶] 博客來Amos水蠟筆5/1特價五折
icon.pngRe: [心得] 新鮮人一些面試分享
icon.png[心得] 蒼の海賊龍 地獄 麒麟25PT
icon.pngRe: [閒聊] (君の名は。雷慎入) 君名二創漫畫翻譯
icon.pngRe: [閒聊] OGN中場影片:失蹤人口局 (英文字幕)
icon.png[問題] 台灣大哥大4G訊號差
icon.png[出售] [全國]全新千尋侘草LED燈, 水草

請輸入看板名稱,例如:Soft_Job站內搜尋

TOP