課程名稱︰應用數學二
課程性質︰物理系必帶
課程教師︰李慶德
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰2018/11/14
考試時限（分鐘）：180分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Do any 5 of the exam problems given below!
1.Consider the differential equations
(a)(e^x siny-2y sinx)dx+(e^x cosy+2cosx)dy=0;
(b)ydx+(2xy-e^(-2y))dy=0
Solve the equations immediately if they are exact equations. Otherwise, try
to solve the equations by first finding an integrating factor that depends
only on x (or only on y).
2.Solve the differential equations:
(a) y'=1/(e^(-y^2/2)-xy)
(b) y'=x^2+y^2+2xy+4x+4y+4
3.Often a differential equation with variable coefficients such as
y''+p(t)y'+q(t)y=0
can be put in a more suitable form for finding a solution by making a
change of independent and/or dependent variables.
(a)Let x=u(t)=∫[αq(t)]^(1/2)dt be the new independent variable in which α
is a constant chosen such that αq(t)>0. Show that if
q'(t)+2p(t)q(t)