- 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
- 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 character
- 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
- 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
- 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
- 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
- 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<# #< $b/<> oe ! . Canceling the exponent
- CHAPTER 2. -
Chapter Two
Section 2.1 1a+b
a,b Based on the direction field, all solutions seem to converge to a specific increasing function. a- b The integrating factor is .a>b oe /$> , and hence C
- 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 so
Math 219, Homework 4
Due date: 30.12.2005, Wednesday Suppose that K > 0, and f (t) is defined as 1 0 if 4n t < 4n + 1 otherwise
f (t) =
where n runs through the set of integers. (a) Determine the Four
Math 219, Homework 3
Due date: 9.12.2005, Friday 1. Consider the initial value problem d2 x dx + + x = u4 (t), dt2 dt y(0) = y (0) = 0
(a) Solve this initial value problem using the Laplace transform.
Math 219, Homework 2
Due date: 23.11.2005, Wednesday This homework concerns two (fictitious) design problems about the solar car "MES e" of the METU Robotics Society, which won the Formula-G trophy in