Math 213
Name:
Midterm Exam
1. [6 points] Find a constant k so that y = x k is a solution to the equation x 2 y 00 = 20y.
2. [6 points] Use guess & check to find one solution to the differential equation e 4x y 0 = y 3 + e 6x .
3. [6 points] Find the general solution to the differential equation y 0 + 4y = e−x .
4. [8 points] Find the general solution to the differential equation y 0 = e x y 2 − y 0
5. [14 points] Water is being drained from the bottom of an inverted conical tank. According to Torricelli’s
law, the volume V of the water left in the tank obeys the differential equation
dV
= −kV 1/3
dt
where k is a constant.
(a) [4 pts] Find the general solution to the above equation.
(b) [10 pts] Suppose the tank holds 1000 liters of water, and that water initially drains from the tank
at a rate of 0.20 L/sec. How long will it take for the tank to drain completely?
6. [12 points (4 pts each)] Consider the following system of linear equations.
x + 4y + 7z = 1
2x + 5y + 2z = 5
5x + ay + bz = 8
(a) Give one example of values of a and b for which this system will have no solutions.
(b) Give one example of values of a and b for which this system will have infinitely many solutions.
(c) Give one example of values of a and b for which this system will have a unique solution.
7. [8 points] Find the inverse of the following matrix.


0
1
5

1
1
2

3
4
2
8. [6 points] Suppose that

2
?
x


A = 
?
4
?


?
−1
?

and

A−1 = 

7
?
3
?
2
1
?

0 ,
1
where each ? indicates an unknown entry. Use this information to find the value of x.
9. [6 points] Use Cramer’s rule to solve for x in terms of a and b.
3x + 5y = a
8x + ay = b

0
8


10. [12 points (6 pts each)] For the following questions, let A =  5

0
2
0
2
0
0
0
0
9
4
5
5
3
3
7
0
4

0
7


4 .

0
3
(a) Compute the determinant of A. You must show your work to receive full credit.
(b) Compute the top-left entry of A−1 . You must show your work to receive full credit.
11. [8 points] Compute the following determinant. You must show your work to receive full credit.
1
4
2
4
1
4
2
8 −2 −6
5
4
1
7
8
7 12. [8 points] Consider the vectors
 
 
1
3
 
 
v1 =  3  ,
v2 =  9  ,
1
5
 
4
 
v3 =  7  ,
2
and
Express the vector w as a linear combination of the vectors v1 , v2 , and v3 .
 
6
 
w = 8.
0