- 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
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- 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
- 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 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 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 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 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 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
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MAT 244 - Ordinary Differential Equations
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Term Test, February 25, 2016
Problem 1: ANS: a)
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b). The equation can be written as
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1
y = (1 +