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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