ch2 - CHAPTER 2 MATHEMATICAL MODELS AND NUMERICAL METHODS...

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68 Chapter 2 0 1 -5 0 5 10 15 t x CHAPTER 2 MATHEMATICAL MODELS AND NUMERICAL METHODS SECTION 2.1 POPULATION MODELS Section 2.1 introduces the first of the two major classes of mathematical models studied in the textbook, and is a prerequisite to the discussion of equilibrium solutions and stability in Section 2.2 . In Problems 1–8 we outline the derivation of the desired particular solution, and then sketch some typical solution curves. 1. Noting that 1 because (0) 2, xx >= we write 11 1; 1 (1 ) 1 dx dt dx dt x x  =− =  −−  ∫∫ ln ln( 1) ln ; 1 t x x xt C C e x = + = (0) 2 implies 2; 2( t x Cx x e == = 22 () 212 t tt e ee . Typical solution curves are shown in the figure on the left below. 0 1 2 3 4 5 -1 0 1 2 3 t
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Section 2.1 69 1 -3 -2 -1 0 1 2 3 t x 2. Noting that 10 because (0) 1, xx <= we write 11 1; 1 0 (10 ) 10 dx dt dx dt x x  =+ =  −−  ∫∫ 10 ln ln(10 ) 10 ln ; 10 t x x xt C C e x = + = 10 1 (0) 1 implies ; 9 (10 ) 9 t x Cx x e == = 10 10 10 10 10 () 91 9 t tt e ee ++ . Typical solution curves are shown in the figure on the right at the bottom of the preceding page. 3. Noting that 1 because (0) 3, >= we write (2 ) (1 )(1 ) 1 1 dx dt dx dt x x =− = +− + 2 1 ln( 1) ln( 1) 2 ln ; 1 t x x C C e x + =−+ = + 2 1 (0) 3 implies ; 2( 1) ( 1) 2 t x x e = + 22 1 1 . Typical solution curves are shown in the figure on the left below. 1 2 3 -2 -1 0 1 2 3 4 t
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70 Chapter 2 0 0.25 0.5 -5 0 5 10 t x 4. Noting that 3 2 because (0) 0, xx <= we write 11 1; 6 (3 2 )(3 2 ) 3 2 3 2 dx dt dx dt x x  =+ =  +− +  ∫∫ 12 1 3 2 ln(3 2 ) ln(3 2 ) 6 ln ; 22 2 3 2 t x x xtC C e x + = 12 (0) 0 implies 3 2 (3 2 ) t x Cx x e == + = () 12 12 12 12 31 33 21 t t t t e e xt e e + + . Typical solution curves are shown in the figure on the right at the bottom of the preceding page. 5. Noting that 5 because (0) 8, >= we write (3 ) ; 1 5 (5 ) 5 dx dt dx dt x x =− = −− 15 ln ln( 5) 15 ln ; 5 t x x C C e x = 15 (0) 8 implies 8/ 3; 3 8( 5) t x x e === 15 15 15 40 40 38 83 t tt e ee . Typical solution curves are shown in the figure on the left below. 0 0.25 0.5 -5 0 5 10 t
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Section 2.1 71 0 0.01 0.02 -10 0 10 20 30 t x 6. Noting that 5 because (0) 2, xx <= we write 11 (3 ) ; (1 5 ) (5 ) 5 dx dt dx dt x x  =− +  −−  ∫∫ 15 ln ln(5 ) 15 ln ; 5 t x x xt C C e x = + = 15 (0) 2 implies 2/3; 3 2(5 ) t x Cx x e === 15 15 15 10 10 () 32 23 t tt e ee == ++ . Typical solution curves are shown in the figure on the right at the bottom of the preceding page. 7. Noting that 7 because (0) 11, >= we write (4 2 8 (7 ) 7 dx dt dx dt x x = 28 ln ln( 7) 28 ln ; 7 t x x C C e x =+ = 28 (0) 11 implies 11/ 4; 4 11( 17) t x x e = 28 28 28 77 77 41 1 1 14 t e . Typical solution curves are shown in the figure on the left below. 0 0.1 -5 0 5 10 15 t
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72 Chapter 2 8. Noting that 13 because (0) 17, xx >= we write 11 7; (9 1 ) (1 3 ) 1 3 dx dt dx dt x x  =− =  −−  ∫∫ 91 ln ln( 13) 91 ln ; 13 t x x xt C C e x −−= + = 91 (0) 17 implies 17/ 4; 4 17( 13) t x Cx x e == = 91 91 91 221 221 () 41 7 1 74 t tt e ee .
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ch2 - CHAPTER 2 MATHEMATICAL MODELS AND NUMERICAL METHODS...

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