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# 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 1 2 cos2 and sin 2 y x y x = = , then 1 2 2sin 2 and 2cos 2 y x y x = − = so 1 1 4cos2 4 y x y ′′ = − = − and 2 2 4sin 2 4 . y x y ′′ = − = − Thus 1 1 4 0 y y ′′+ = and 2 2 4 0. y y ′′ + = 4. If 3 3 1 2 and x x y e y e = = , then 3 3 1 2 3 and 3 x x y e y e = = − so 3 1 1 9 9 x y e y ′′ = = and 3 2 2 9 9 . x y e y ′′ = = 5. If x x y e e = , then x x y e e ′ = + so ( ) ( ) 2 . x x x x x y y e e e e e ′ − = + = Thus 2 . x y y e ′ = + 6. If 2 2 1 2 and x x y e y x e = = , then 2 2 2 2 1 1 2 2 , 4 , 2 , x x x x y e y e y e xe ′′ = − = = and 2 2 2 4 4 . x x y e xe ′′ = − + Hence ( ) ( ) ( ) 2 2 2 1 1 1 4 4 4 4 2 4 0 x x x y y y e e e ′′ + + = + + = and ( ) ( ) ( ) 2 2 2 2 2 2 2 2 4 4 4 4 4 2 4 0. x x x x x y y y e xe e xe x e ′′ + + = + + + =

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2 Chapter 1 8. If 1 2 cos cos2 and sin cos2 , y x x y x x = = then 1 sin 2sin2 , y x x ′ = − + 1 cos 4cos2 , y x x ′′= − + and 2 2 cos 2sin2 , sin 4cos2 . y x x y x x ′′ = + = − + Hence ( ) ( ) 1 1 cos 4cos2 cos cos2 3cos2 y y x x x x x ′′+ = + + = and ( ) ( ) 2 2 sin 4cos2 sin cos2 3cos2 . y y x x x x x ′′ + = + + = 11. If 2 1 y y x = = then 3 4 2 and 6 , y x y x ′′ = − = so ( ) ( ) ( ) 2 2 4 3 2 5 4 6 5 2 4 0. x y x y y x x x x x ′′ + + = + + = If 2 2 ln y y x x = = then 3 3 4 4 2 ln and 5 6 ln , y x x x y x x x ′′ = = − + so ( ) ( ) ( ) ( ) ( ) 2 2 4 4 3 3 2 2 2 2 2 2 5 4 5 6 ln 5 2 ln 4 ln 5 5 6 10 4 ln 0. x y x y y x x x x x x x x x x x x x x x x ′′ + + = + + + = + + + = 13. Substitution of rx y e = into 3 2 y y ′ = gives the equation 3 2 rx rx r e e = that simplifies to 3 2. r = Thus r = 2/3. 14. Substitution of rx y e = into 4 y y ′′ = gives the equation 2 4 rx rx r e e = that simplifies to 2 4 1. r = Thus 1/ 2. r = ± 15. Substitution of rx y e = into 2 0 y y y ′′ + = gives the equation 2 2 0 rx rx rx r e r e e + = that simplifies to 2 2 ( 2)( 1) 0. r r r r + = + = Thus r = –2 or r = 1. 16. Substitution of rx y e = into 3 3 4 0 y y y ′′ + = gives the equation 2 3 3 4 0 rx rx rx r e r e e + = that simplifies to 2 3 3 4 0. r r + = The quadratic formula then gives the solutions ( ) 3 57 /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.
Section 1.1

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

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