课程名称︰应用数学二 课程性质︰必修 课程教师︰李庆德 开课学院：理学院 开课系所︰物理系 考试日期（年月日）︰2018/10/31 试题 : (Do any 4 of the exam problems given below) 1. Find the general solution of the DE y''+2y'+5y=16exp(x)+sin(2x) 2. Find the general solution of the DE y''+4y=sec(x) 3. Solve the following non-homogeneous Euler equation y''x^2-4xy'+6y=x^2 4. Consider the differential equation xy''-(x+N)y'+Ny=g(x), where N is a nonnegative integer. One reason this differential equation is interesting is that the associated homogeneous equation has an exponential solution and a polynomial solution. (a) Verify that the differential equation can be written as the factored form (xD-N)(D-1)y=g(x), where D denotes the differential with respect to x. (b) Show that the solution to the differential equation can be found by solving the following two first order equations: (xD-N)u=g(x), (D-1)y=u(x). (c) Solve the differential equation for N=3 and g(x)=x^2. 5. Suppose y1 and y2 are particular solutions of y'''-4y''+3y'=f(x), where f(x) is a non-zero differentiable function. Which of the following are also solutions to the above differential equation, and why? (a) y(x)=2+y1(x) (b) y(x)=y1(x)-y2(x)+1 (c) y(x)=1+exp(x)+exp(2x) (d) y(x)=exp(x)-y2(x) (e) y(x)=2y2(x)-y1(x)-1 --

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