課程名稱︰常微分方程導論
課程性質︰數學系大二必修
課程教師︰陳俊全教授
開課學院：理學院
開課系所︰數學系
考試日期︰2017年11月10日(五)
考試時限：08:10-09:00，共計50分鐘
試題 :
106 Introduction to Ordinary Differential Equations
Quiz 1
Date: 2017.11.10
Write down all details in your argument as possible as you can.
Part I:
1. (20%) Solve the following equations.
(a) 2y' + xy = e^(-x^2/4).
(b) y cos(x) + 2xe^y +(sin(x)+x^2e^y-1)y' = 0.
2. (20%) Solve the Bernoulli equation y' - 2y/x = -x^2y^2.
3. (20%) Solve (-3x+y+6)dx + (x+y+2)dy = 0.
4. (20%) Consider a pond that initially contains 10 million gal of fresh water.
Water containing an undersirable chemical flows into the pond at the rate of
5 million gal/yr, and the mixture in the pond flows out at the same rate.
The concentration γ(t) of chemical in the incoming water varies periodical-
ly with time according to the expression γ(t) = 2 + sin(2t) g/gal. Determi-
ne the amount of chemical in the pond at any time.
Part II: Choose one of following two problems.
5. (20%) Recall the Picard iteration, which is a method of successive approxim-
ations to approximate the solution of the initial value problem
y' = f(t,y), (1)
y(0) = y0,
defined by ψ(t) = y0 and
0 t
ψ(t) = y0 + ∫f(s,ψ(s))ds, n 屬於 |N (2)
n 0 n-1
Consider f(t,y) = -(sin(t))y + 1 and y0 = 0.
(a) Calculate ψ(t), ψ(t), ψ(t).
1 2 3
(b) Estimate the error: |ψ(t) - ψ(t)|, where ψ(t) is the solution to (1).
3
6. (20%) Consider the initial value problem
y' = M^4 - y^4, (3)
y(0) = y0,
where M > 0 is a given constnant. Suppose 0 < y0 < M. Do not solve the equa-
tion directly, use mathematical analysis to solve following problems.
(a) Prove that y(t) is bounded and increasing.
(b) Find lim y(t) and justify your result.
t→∞