By Carlos A. Smith

ISBN-10: 1439850887

ISBN-13: 9781439850886

IntroductionAn Introductory ExampleModelingDifferential EquationsForcing FunctionsBook ObjectivesObjects in a Gravitational FieldAn instance Antidifferentiation: process for fixing First-Order usual Differential EquationsBack to part 2-1Another ExampleSeparation of Variables: method for fixing First-Order usual Differential Equations again to part 2-5Equations, Unknowns, and levels ofRead more...

summary: IntroductionAn Introductory ExampleModelingDifferential EquationsForcing FunctionsBook ObjectivesObjects in a Gravitational FieldAn instance Antidifferentiation: procedure for fixing First-Order usual Differential EquationsBack to part 2-1Another ExampleSeparation of Variables: strategy for fixing First-Order traditional Differential Equations again to part 2-5Equations, Unknowns, and levels of FreedomClassical options of normal Linear Differential EquationsExamples of Differential EquationsDefinition of a Linear Differential EquationIntegrating issue MethodCharacteristic Equation

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IntroductionAn Introductory ExampleModelingDifferential EquationsForcing FunctionsBook ObjectivesObjects in a Gravitational FieldAn instance Antidifferentiation: process for fixing First-Order usual 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 recommendations of normal Linear Differential EquationsExamples of Differential EquationsDefinition of a Linear Differential EquationIntegrating issue MethodCharacteristic Equation.

**Extra resources for A First Course in Differential Equations, Modeling, and Simulation**

**Example text**

The forcing functions were all different, and therefore, the particular solutions were also different. 11 The following differential equation describes an undamped mass-spring system: x″ + 16x = 4 sin ωt Obtain its solution. We start by finding the solution to the corresponding homogeneous equation, xH′′ + 16 xH = 0 Assuming xH = ert, we get r2 + 16 = 0 ⇒ r1 = 4i ; r2 = –4i And using the previous treatment, xH = C1 cos 4t + C2 sin 4t Note that the frequency of this homogeneous response or natural response is 4 radians/time.

Most of the important information about the system response can be obtained from these roots. Classical Solutions of Ordinary Linear Differential Equations 39 The relevant questions about the response are the following: • Is the response stable? That is, will the response remain bounded when forced by a bounded input? • Is the response monotonic or oscillatory? • If monotonic and stable, how long will it take for the transients to die out? • If oscillatory, what is the period of oscillation and how long will it take for the oscillations to die out?

16) i=1 or, in this example, y = Cn e rnt + Cn−1 e rn−1t + … + C1 e r1t 38 A First Course in Differential Equations, Modeling, and Simulation The constants Cn, Cn–1, … , and C1 are evaluated using the initial conditions. This method is fairly simple; the most difficult step is obtaining the roots of the characteristic equation. 5 Obtain the solution for the following differential equation: y″ + y ′ – 12y = 0 with y ′(0) = 0 y(0) = 3 Assuming the solution y = ert and following the procedure previously shown, we obtain the characteristic equation, dy d 2y = r e rt ; = r 2 e rt dt dt 2 r2 ert + r ert – 12ert = 0 r2 + r – 12 = 0 Using the quadratic equation to find the roots, r1, r2 = 3, –4.

### A First Course in Differential Equations, Modeling, and Simulation by Carlos A. Smith

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