- CHAPTER 1. -
Chapter One
Section 1.1 1.
For C "& , the slopes are negative, and hence the solutions decrease. For C "& , the slopes are positive, and hence the solutions increase. The equilibrium solution appears to be Ca>b oe "& , to which all other so
- CHAPTER 11. -
Chapter Eleven
Section 11.1 1. Since the right hand sides of the ODE and the boundary conditions are all zero, the boundary value problem is homogeneous. 3. The right hand side of the ODE is nonzero. Therefore the boundary value problem is
- CHAPTER 9. -
Chapter Nine
Section 9.1 2a+b Setting x oe 0 /<> results in the algebraic equations OE &< $
For a nonzero solution, we must have ./>aA < Ib oe <# ' < ) oe ! . The roots of the characteristic equation are <" oe # and <# oe % . For < oe #, th
- CHAPTER 8. -
Chapter Eight
Section 8.1 2. The Euler formula for this problem is C8" oe C8 2^& >8 $C8 , C8" oe C8 &82# $2 C8 ,
in which >8 oe >! 82 Since >! oe ! , we can also write
a+b. Euler method with 2 oe !& >8 C8 8oe# !" "&*)! 8oe% !# "#*#) 8oe' !$
- CHAPTER 6. -
Chapter Six
Section 6.1 3.
The function 0 a>b is continuous. 4.
The function 0 a>b has a jump discontinuity at > oe " . 7. Integration is a linear operation. It follows that (
E !
-9=2 ,> /=> .> oe
" E ,> => " E ,> => ( / / .> ( / / .> # !
- CHAPTER 5. -
Chapter Five
Section 5.1 1. Apply the ratio test : lim aB $b8" k a B $b 8 k
Hence the series converges absolutely for kB $k " . The radius of convergence is 3 oe " . The series diverges for B oe # and B oe % , since the n-th term does not a
- CHAPTER 4. -
Chapter Four
Section 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function 1a>b oe > , are continuous everywhere. Hence solutions are valid on the entire real line. 3. Writing the equation in standa
- CHAPTER 3. -
Chapter Three
Section 3.1 1. Let C oe /<> , so that C w oe < /<> and C ww oe < /<> . Direct substitution into the differential equation yields a<# #< $b/<> oe ! . Canceling the exponential, the characteristic equation is <# #< $ oe ! The ro
- CHAPTER 2. -
Chapter Two
Section 2.1 1a+b
a,b Based on the direction field, all solutions seem to converge to a specific increasing function. a- b The integrating factor is .a>b oe /$> , and hence Ca>b oe >$ "* /#> - /$> It follows that all solutions co
Program Level Assessment SPRING 2017
Job opportunity surveys have consistently shown a high demand for those who can use mathematics to
model complex processes and make informed predictions of the future. A student should be able to
demonstrate his or her