课程名称︰工程数学-微分方程(一)
课程性质︰电机工程学系大二共同必修
课程教师︰丁建均
开课学院：电机资讯学院
开课系所︰电机工程学系
考试日期（年月日）︰2019/11/6
考试时限（分钟）：10:20-12:10 (110分钟)
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
1.Solve the following DE and give the corresponding largest interval for the
solution (if the explicit solution exists, please do not write as the
implicit solution, On the other hand, you may have to express the solution
in terms of an integral-defined function for some problem(s).)
(a) x (1+y^2)^(1/2) dx = y (1+x^2)^(1/2) dy (5%)
(b) dy/dx = exp(sqrt(x))/y , y(1) = 4 (5%)
(c) dr/dΘ + rsecΘ = cosΘ (5%)
(d) dy/dx+ exp(x)*y = 1 , y(0) = 1 (5%)
(e) dx/dy - xsin(y) = 2sin(y) , x(π/2) = 1 (5%)
2.Solve the exact equation:
dx/dy = -(4y^2 + 6xy)/(3y^2+2x) (5%)
3.Solve the following differential equations by using appropriate substitution
(a) dy/dx = (x + y + 1)^2 (5%)
(b) x(dy/dx) = y + sqrt(x^2-y^2) , x>0 (5%)
4.The mathematical model for the velocity v of a chain slipping off the edge
of horizontal platform is xv(dv/dx) + v^2 = gx , where x denote the
length of the chain overlies the table, v is its velocity, and g is the
gravity.
(a) Please rewrite the equation in the form of M(x,v)dx + N(x,v)dv
= 0 and then solve as a modified exact equation with an appropriate
integrating factor. (5%)
(b) Please rewrite the equation in the form of dv/dx + P(x)v =
f(x)v'' and then solve as a Bernoulli equation with an appropriate
substitution. (5%)
5.Suppose that in the temperature in a refrigerator is -7℃ and the temperatur
-e outside the refrigerator is 25℃. If we take a drinl out from the refrige
-rator, after 1 hour, the temperature of the drink is 17℃.
(a) Express how the temperature of the drink varies with time by a DE.(5%)
(b) Suppose that, after 1 hour, the temperature of the drink is 17℃.
How long will the temperature of the drink be 24℃? (5%)
6.Suppose that (x-2)y''(x) + xy'(x) + [(x+2)/(x+1)]y(x) = 0, y(Xo) = 0,
y'(Xo) = 1. In what conditions the DE may not have a unique solution(Show
the solutions in terms of Xo). (5%)
7.Solve the general solutions of the following DEs:
(a) (x+2)y''(x) - y'(x) = 0, y(x) = 1 is one of the solutions (5%)
(b) sin(x)y''(x) - cos(x)y'(x) = 0, y(x) = 1 is one of the solutions (5%)
8.For the following higher-order linear differential equations:
(a) Use the Auxiliary Equation Approach to determine the general
solution of y'''(x) - 6y''(x) + 11y'(x) - 6y(x) = 0. (7%)
(b) Use the Superposition Approach to determine the general solution of
y''(x) - 3y'(x) + 2y(x) = exp(3x). (7%)
(c) Use the Annihilator Approach to determine the general solution of
y''(x) - 3y'(x) + 2y(x) = exp(3x). (7%)
(d) Are these three general solutions you found for (a), (b), and (c)
the same? If yes, verify. If no, describe the reason. (4%)