課程名稱︰應用數學二
課程性質︰必修
課程教師︰李慶德
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2018/10/31
試題 :
(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