ch1 - CHAPTER 1 FIRST-ORDER DIFFERENTIAL EQUATIONS SECTION...

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Section 1.1 1 CHAPTER 1 FIRST-ORDER DIFFERENTIAL EQUATIONS SECTION 1.1 DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELS The main purpose of Section 1.1 is simply to introduce the basic notation and terminology of differential equations, and to show the student what is meant by a solution of a differential equation. Also, the use of differential equations in the mathematical modeling of real-world phenomena is outlined. Problems 1–12 are routine verifications by direct substitution of the suggested solutions into the given differential equations. We include here just some typical examples of such verifications. 3. If 12 cos2 and sin 2 yx == , then 2sin 2 and 2cos 2 y x ′′ =− = so 11 4cos2 4 y ′′=− and 22 4sin2 4 . y Thus 40 yy += and . 4. If 33 and xx ye , then 3a n d 3 y e so 3 99 x y and 3 . x y 5. If ye e , then yee =+ so () 2. x ee e −− −= + = Thus x e 6. If and y x e , then 2 2 2 2, 4, 2 , x x y eyey e x e = = and 2 44 . x e + Hence ( ) ( ) 2 111 4 444 24 0 x yyy e e e ++ = + + = and ( ) ( ) 2 222 4 4 4 2 4 0 . x e x e e x e x e = + + + =
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2 Chapter 1 8. If 12 cos cos2 and sin cos2 , yx x x =− then 1 sin 2sin2 , yxx ′=− + 1 cos 4cos2 , ′′=− + and 22 cos sin 4cos2 . x y x x ′′ =+ = + Hence () ( ) 11 cos 4cos2 cos 3cos2 yy x x x x x += + + = and ( ) sin sin 3cos2 . x x x x x + + = 11. If 2 1 yy x == then 34 2a n d 6 , y x −− = so ()( ) ( ) 4 3 2 54 6 5 2 4 0 . xy y x x x x x ++ = + + = If 2 2 ln yy x x then 33 44 2l n a n d 5 6l n , x x y x x x = + so ( ) ( ) ( ) 4 4 3 3 2 2 56 l n 5 2 l n 4 l n 55 61 04 l n0 . x yyx x xxx x xx x + + = −+ + + + = 13. Substitution of rx ye = into 32 ′ = gives the equation 3 2 rx rx re e = that simplifies to . r = Thus r = 2/3. 14. Substitution of rx = into 4 ′′ = gives the equation 2 4 rx rx e = that simplifies to 2 41 . r = Thus 1/2. r 15. Substitution of rx = into 20 y +− = gives the equation 2 rx rx rx e +−= that simplifies to 2 2( 2 ) ( 1 )0 . rr r r +− = + − = Thus r = –2 or r = 1. 16. Substitution of rx = into 334 0 y gives the equation 2 4 0 rx rx rx e +− = that simplifies to 2 40 . The quadratic formula then gives the solutions 35 7 / 6 . r =−± The verifications of the suggested solutions in Problems 17–26 are similar to those in Problems 1–12. We illustrate the determination of the value of C only in some typical cases. However, we illustrate typical solution curves for each of these problems.
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Section 1.1 3 -5 0 5 -5 0 5 x y (0,3) -10 -5 0 5 10 -20 0 20 x (0,10) 17. C = 2 18. C = 3 19. If ( ) 1 x yx Ce =− then y (0) = 5 gives C – 1 = 5, so C = 6. The figure is on the left below. 20. If ( ) 1 x x =+ then y (0) = 10 gives C – 1 = 10, so C = 11. The figure is on the right above. 21. C = 7. The figure is on the left at the top of the next page. -5 0 5 -5 0 5 x (0,2) -5 0 5 -10 -5 0 5 10 x (0,5)
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4 Chapter 1 -20 -10 0 10 20 -5 0 5 x y (0,0) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -30 -20 -10 0 10 20 30 x (1,1) 22. If () l n ( ) yx x C =+ then y (0) = 0 gives ln C = 0, so C = 1. The figure is on the right above.
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This note was uploaded on 04/08/2008 for the course MATH 374 taught by Professor Zhu during the Spring '08 term at Western Michigan.

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ch1 - CHAPTER 1 FIRST-ORDER DIFFERENTIAL EQUATIONS SECTION...

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