課程名稱︰微分方程
課程性質︰電機系大二必修
課程教師︰丁建鈞
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰96/11/21
考試時限（分鐘）：130
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Solving the following 1st order differential equations
(show the explicit solutions)
￣￣￣￣￣￣￣￣￣
(a)(7%) dy/dx = y + 3
(b)(8%) dy/dx = y + 3x
(c)(10%) dy/dx = (y^2 - 9)/x
2. In 2003, there are 30 pandas in an isolated zoo, including five pandas
inder one year old. Assume the growth rate is proportional to the panda
population.
(a)(5%)List the IVP and find the year (20??) of the panda population over 100.
(b)(5%)List the IVP under the carrying capacity of the zoo environment of
150 pandas, and find the number of pandas corresponding to the point
of inflection of the solution in (b)
(c)(5%)Using Bernoulli's Equation to solve the DE in (b), and find the
year (20??) of the panda population over 100.
3.(10%)Solve the given initial-value problem
dx 2
——— = ——————————— ；
dy ( 10 - 6y + e^(-3x)
5
y(0) = ——— 。
3
4.(a)(5%)Define a linear operator L3 = D^3 - 6D^2 + 12D - 8.
Please determine the complementary function yc(x), a particular
solution yp(x) and the general solution for L3(y) = 4e^(2x)
(b)(5%)In the yc(x), if the e^2x term is removed, please find the
second-order linear differential equation L2(y) = 0 whose
complementary function is the linear combination of the left terms
and coefficient of D^2 is 1.
(c)(5%)Find the general solution and a particular solution for
L2(y) = 4e^2x
(d)(5%)The solution interval of differential equation is defined as the
x interval where the IVP has a unique solution for every x point.
The singular point of a differential equation is defined as the
x point where the IVP has no unique solution. Please find the
solution intervals and singular points for L3(y) = 4e^2x and
L2(y) = 4e^2x. Please note the interval number and point number
may not be one.
(e)(5%)Is the statement, " Every solution for L2(y) = 4e^2x belongs to
L3(y) = 4e^2x. " correct or not ? Give the reason at the same
time. How about the homogeneous case ?
5. Find the general solution of
4 ,, 3 , 2
(a)(7%) X Y + X Y - 4 X Y = 0
4 ,, 3 , 2
(b)(8%) X Y + X Y - 4 X Y = 1
2 2
d Y – g R
6. Plese solve the equation ———— = —————
2 2
d t y
where y is the distance with the unit of length.
(a)(5%)Express your solution that relates y and the velocity v = y' .
The initial conditions are y(0) = R, y'(0) = v0 .
(b)(5%)If we would like to have the value of y' equal to zero for
y ---> ∞, what is the value of v0.?
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