Chapter7diffeq - CHAPTER 7 Nonlinear Systems of...

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682 CHAPTER 7 Nonlinear Systems of Differential Equations 7.1 Nonlinear Systems Review of Classifications 1. 2 sin x x ty y x y t γ ′ = + ′ = + + Dependent variables: , x y Parameter: γ Nonautonomous linear system Nonhomogeneous ( sin ) t γ 2. 3 4 2 sin u u u t υ υ ′ = + ′ = − + Dependent variables: , u υ Parameters: none Nonautonomous linear system Nonhomogeneous (sin ) t 3. 1 2 2 1 = =—sin x x x x κ Dependent variables: 1 2 , x x Parameter: κ Autonomous nonlinear 1 (sin ) x system 4. sin p q q pq t ′ = ′ = Dependent variables: p , q Parameters: none Nonautonomous nonlinear ( ) pq system 5. S rSI I rSI I R I γ γ ′ = − ′ = − ′ = Dependent variables: R , S , I Parameters: , r γ Autonomous nonlinear ( ) SI system Verification Review 6. = x x y y ′ = Substituting t x y e = = into the two differential equations, we get = = t t t t e e e e 7. x y y x ′ = ′ = − Substituting sin x t = and sin y t = into the two differential equations, we get cos cos sin sin t t t t = = − 8, 9. Use direct substitution, as in Problems 6 & 7.
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