By Carlos A. Smith
IntroductionAn Introductory ExampleModelingDifferential EquationsForcing FunctionsBook ObjectivesObjects in a Gravitational FieldAn instance Antidifferentiation: procedure for fixing First-Order traditional Differential EquationsBack to part 2-1Another ExampleSeparation of Variables: procedure for fixing First-Order traditional Differential Equations again to part 2-5Equations, Unknowns, and levels of FreedomClassical ideas of normal Linear Differential EquationsExamples of Differential EquationsDefinition of a Linear Differential EquationIntegrating issue MethodCharacteristic Equation. Read more...
summary: IntroductionAn Introductory ExampleModelingDifferential EquationsForcing FunctionsBook ObjectivesObjects in a Gravitational FieldAn instance Antidifferentiation: approach for fixing First-Order traditional Differential EquationsBack to part 2-1Another ExampleSeparation of Variables: strategy for fixing First-Order usual Differential Equations again to part 2-5Equations, Unknowns, and levels of FreedomClassical strategies of normal Linear Differential EquationsExamples of Differential EquationsDefinition of a Linear Differential EquationIntegrating issue MethodCharacteristic Equation
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Extra info for A First Course in Differential Equations, Modeling, and Simulation
Then, dT hA γ + T= e − βt dt mC p mC p dT + a T = b e − βt dt where a = (hA/mCp) ; b = (γ/mCp) ; P(t) = a, and Q(t) = b e–βt. 11. 13. 11. 4 Characteristic Equation The characteristic equation method is the technique for solving homogeneous nth-order linear differential equations with constant (time invariant) coefficients. This method provides the solution for a constant-coefficient linear homogeneous differential equation. 14) n − 1 n n− 1 dt dt dt where an, an–1, a1, and a0 are constant coefficients.
1. 15). • If any part of the particular solution is a solution of the homogeneous equation, multiply that particular solution by the independent variable. 10). 5 except that this time it has a forcing function (it is a nonhomogeneous equation). 10, y = yH + yP The corresponding homogeneous equation is yH′′ + yH′ − 12yH = 0 Using the characteristic equation method, assume yH = ert and after substituting into the homogeneous equation, r2 + r – 12 = 0 (r + 4)(r – 3) = 0 ⇒ r1 = 3 ; r2 = –4 One root, r1, is located in the positive real axis, and the other root, r2, is in the negative real axis, thus indicating an unstable response.
17 indicates that there are two unknowns, vy and ΣFy. This book shows how to solve this differential equation for vy if everything is known; therefore, we need an equation for ΣFy. Thus, the procedure indicates what is the next equation needed. We will continue using this procedure throughout the book and strongly encourage the reader to do the same. 22) Thus, to solve any set of equations, DoF = 0! , DoF ≠ 0), then the engineer/designer has the “freedom” to select/specify the value(s) of any of the unknown(s) necessary to make DoF = 0.