[試題] 97上 陳逸聰 化學數學 期中考

[candy79729 (candy79729)]

課程名稱︰化學數學
課程性質︰化學系系必修
課程教師︰陳逸聰
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰ 97 / 11 / 11
考試時限（分鐘）： 10 : 20 ~ 13 : 00 ，共 160 分鐘
是否需發放獎勵金： 是

試題 :





1. If a fossilized tree is claimed to be 4000 years old, what should be its
   C14 content expressed as a percent of the ratio of C14 to C12 in a living
   organism? (6%)






2. Test for exactness. If exact, solve. If not exact, find an integrating
   factor and determind the corresponding particular solution. (10%)

     ( cos xy + x/y ) dx + ( 1 + x/y cos xy ) dy = 0






3. (a) Solve the following Bernouli equation. (5%)

         y' = Ay - By^2

   (b) Suppose that the population y(t) of a certain kind of fish is given by
       the equation of (a), and fish are caught at a rate Hy propotional to y.
       Solve this so-called Schaefer model. Find the equilibrium solutions y1
       and y2 (>0) when H<A. The expression Y = Hy2 is called the equilibrium
       harvest or sustainadle yield corresponding to H. Why? (5%)






4. Find the frequency of oscillation of a pendulum of length L, neglecting air
   resistance and the weight of the rod, and assuming θ to be so small that
   sin θ practically equals θ.

　　　 　 ┬─
　　　　　│\
　　　　　│ \ L
　　　　　│  \
　　　　　│θ \
　　　　　│　 ○Body of mass m






5. (a) Find the solution for a forced oscillator of the following equation.
       (6%)

        y" + 25y = 99 cos 4.9t, y(0) = 2, y'(0) = 0

   (b) How does the graph of the solution in (a) charge if you change
       (i) y(0)
       (ii) the frequency of the driving force? (4%)






6. Are cosh x = ( e^x + e^-x ) / 2 and sinh x = ( e^x - e^-x ) / 2 linearly
   independent? Why? (6%)






7. Solve the nonhomogeneous ODE by variation of parameters or undermined
   coefficients. Give a general solution. (10%)
   Hint: To find y1, y2, set y = ux^-1/2

   [ x^2*D^2 + xD + (x^2 - 1/4 ) I ] y = x^3/2 sin x






8. Find the current in the RCL-circuit when L=0.4H, R=40Ω, C=10^-4 F, and
   E(t) = 220 sin 314t V (50 cycles/sec) (8%)






9. Find the steady-state solution of the damped forced oscillating system when
   m=4, c=4, k=17 and the driving force is 202 cos 3t.

　   　┬
　   　ξ
     k ξ  Spring
       ξ
     m ■  Mass
       │
     c ︼  Dashpot





10. Solve the following initial value problem. Sketch or graph the solution.
    (10%)

      y" + 6y' + 18y = 0    y(0) = 5, y'(0) = 21





11. Solve the following initial value problem. (8%)

    x^2 y" + 15xy' + 49y = 0    y(1) = 2, y'(1) = -11





12. Find the solutions of y'" - 3y" + 3y' - y = 0 and prove that your
    solutions are linearly independent. (6%)





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◆ From: 118.166.10.7
推 abacada     :時間沒有延長寫得完嗎? @@
推 eminemk10   :抱歉 我是他校的化學系學生 可以請問化數的教科書是?
推 abacada     :我之前修課是用Kri-開頭那本厚厚的工程數學教科書
推 abacada     :不知道現在是? @@
推 Julibea     :這次用Erwin Kreyszig的書
→ Julibea     :Advanced Engineering Mathematics 9e
→ eminemk10   :Orz 感恩
※ 編輯: candy79729      來自: 118.166.12.194       (12/14 19:33)
→ candy79729  :囧我忘了 有延長到1點......
推 abacada     :哈  還是老樣子  不過改版了XD  我用8e
推 Julibea     :上課也是有人拿8e來用啦  應該沒差多少?
推 abacada     :應該是吧