Solutions to Midterm Exam 1
Math 2250-3, Fall 2007
Instructions: No calculators of any kind are permitted for use on the test, nor is scratch paper allowed.
You have 50 minutes in which to answer all 8 questions. The value of each question as a percentage of the
value of the entire test is indicated next to each question number (this is also approximately the percentage
of 50 minutes each question should require: 5% = 2.5 minutes). Please show all your work in the space
provided below each question.
1. (5%) If the first-order ODE
dx
dt =f(x, t)
is autonomous, what does this imply about the function f?
fis autonomous if and only if f(x, t) = f(x), i.e., fis a function of xonly.
2. (15%) Solve the initial value problem
dy
dx =y1/2
x+ 1 , y (0) = 0.
This is separable!
Separate variables: y−1/2dy =dx
x+ 1
Integrate: 2y1/2= ln(x+ 1) + C
Solve for y:y(x) = (ln(x+ 1) + C)2
4
Solve for C: 0 = y(0) = (ln(1) + C)2
4=C2
4=> C = 0
Particular Solution: y(x) = (ln(x+ 1))2
4
3. (15%) Solve the initial value problem
dy
dx =−2y
x+cos(x)
x2, y(π) = 1.
This is linear!
Find Pand Q:P(x) = 2
x, Q(x) = cos(x)
x2
Integrating factor: eRP(x)dx = e2Rdx/x = e2 ln(x)=x2
Integral with Q:ZeRP(x)dxQ(x)dx =Zx2cos(x)
x2dx =Zcos(x)dx = sin(x) + C
General solution: y(x) = e−RP(x)dx ZeRP(x)dx Q(x)dx +C=sin(x) + C
x2
Solve for C: 1 = y(π) = sin(π) + C
π2=C
π2⇒C=π2
Particular Solution: y(x) = sin(x) + π2
x2
