课程名称︰工程数学二 课程性质︰系必修 课程教师︰王胜仕 开课学院：工学院 开课系所︰化工系 考试日期（年月日）︰2011年4月29日 考试时限（分钟）：140分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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与P1,P2,P3间的关系?) (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. --

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