108-3: Engineering Mathematics (I)
Exam-2
Time: 2020-07-18 Tue. 8:10AM-10:40AM (No extension)
Contact
Name
E-mail
Location
Extension
Lecturer
Wenson Chang
wenson@ee.ncku.edu.tw
92A31
62392
TA1
TA2
程浩偉
king295705@gmail.com
92912
62400-1612
李庚道
philiplovemaster@gmail.com
92912
62400-1612
Problem 1(15%):
Solve y 00 + 9y = −4x sin(3x)
Sol:
Check Example 2.22.
Problem 2(20%):
Consider the differential equation y 00 + Ay 0 + By = 0.
1. (a)(5%) Write down it’s characteristic equation.
2. (b)(15%) Write down it’s real solution when it has complex roots.
Sol:
Check Section 2.4.3.
Problem 3(15%):
Solve the differential equation.
x2 y 00 − 3xy 0 + 4y = 0; y(1) = 4; y 0 (1) = 5.
Sol:
Check Problem 11 of the exercise for Section 2.6.
Problem 4(20%):
Prove the Theorem of Wronskian Test, which is listed as follows.
Let y1 and y2 be solutions of y 00 + p(x)y 0 + q(x)y = 0 on an open interval I. Then,
(1)(10%) Either W (x) = 0 for all x in I, or W (x) 6= 0 for all x in I.
(2)(15%) y1 and y2 are linearly independent on I if and only if W (x) 6= 0 on I.
Sol: Check pages 5-6 of the lecture notes for Ch2.
Problem 5(30%):
Solve x2 y 00 − 3xy 0 + 5y = 5 ln(−x) + 1 for x < 0
1. (a)(15%) Find the homogeneous solution, i.e. yh .
2. (b)(15%) Find the general solution, i.e. y = yh + yp , where yp is the particular solution.
Sol:
1
1. (a)(15%) This is Euler’s equation. Following the procedure in Ch2.5 gives yh = x2 [c1 cos(ln(−x)) + c2 sin(ln(−x))]
or you can write it as yh = x2 [c1 cos(ln(|x|)) + c2 sin(ln(|x|))].
2. (b)(15%) Similar to Example 2.23 y = x2 [c1 cos(ln(−x)) + c2 sin(ln(−x))] + ln(−x) + 1. or
y = x2 [c1 cos(ln(|x|)) + c2 sin(ln(|x|))] + ln |x| + 1.
2