Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 21
... initial value problem the mass is initially at rest . Two initial conditions are necessary since the differential equation involves the second derivative in time . To solve this initial value problem , the arbitrary constants c1 and c2 ...
... initial value problem the mass is initially at rest . Two initial conditions are necessary since the differential equation involves the second derivative in time . To solve this initial value problem , the arbitrary constants c1 and c2 ...
Page 42
... initial conditions can be satisfied . Show that this solution does not approximate very well the solution to ( a ) for all time . 13.5 . Assume that c2 > 4mk . ( a ) Solve the initial value problem , that is , at t = 0 , x = ( b ) Take ...
... initial conditions can be satisfied . Show that this solution does not approximate very well the solution to ( a ) for all time . 13.5 . Assume that c2 > 4mk . ( a ) Solve the initial value problem , that is , at t = 0 , x = ( b ) Take ...
Page 75
... initial conditions . Thus the entire qualitative behavior of the solution can be determined by analyzing the phase plane . EXERCISES 21.1 . Suppose the motion of a mass m was described by the nonlinear differential equation m ( d2x ...
... initial conditions . Thus the entire qualitative behavior of the solution can be determined by analyzing the phase plane . EXERCISES 21.1 . Suppose the motion of a mass m was described by the nonlinear differential equation m ( d2x ...
Contents
Mechanical Vibrations 1 | 3 |
NEWTONS LAW | 4 |
NEWTONS LAW AS APPLIED TO A SPRINGMASS SYSTEM | 6 |
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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analyze approximation Assume c₁ c₂ calculated conservation of cars Consider corresponding d2x dt2 de/dt density wave velocity depends derived determine dq/dp dx dt dx/dt equilibrium population equilibrium position equilibrium solution example exercise exponential Figure flow-density force formula friction function growth rate highway increases initial conditions initial density initial traffic density initial value problem integral intersect isoclines logistic equation mass mathematical model maximum method of characteristics motion moving Newton's nonlinear pendulum number of cars observer occurs ordinary differential equation oscillation P/Pmax P₁ partial differential equation period phase plane Pmax potential energy problem qualitative region result sharks shock velocity Show shown in Fig simple harmonic motion Sketch the solution sketched in Fig slope solve species spring spring-mass system stable straight line Suppose Taylor series tion traffic flow traffic light trajectories Umax Umaxt unstable equilibrium variables velocity-density x₁ yields zero др