Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 62
... dx / dt , dx d2x m dt dt2 di di2 = − f ( x ) dx . The left - hand side is an exact derivative , since Thus , After multiplying by dt , m di 1 di [ 2 ( d / zx ) 2 ] = 1 dt dx d2x dt dt2 d [ = ( dx ) ' ] = − f ( x ) dx . dt 2 1 dt md [ 1⁄2 ...
... dx / dt , dx d2x m dt dt2 di di2 = − f ( x ) dx . The left - hand side is an exact derivative , since Thus , After multiplying by dt , m di 1 di [ 2 ( d / zx ) 2 ] = 1 dt dx d2x dt dt2 d [ = ( dx ) ' ] = − f ( x ) dx . dt 2 1 dt md [ 1⁄2 ...
Page 68
... dx / dt vs. x space , for each value of E : dx dt 293 X Figure 20-1 Typical energy curve . For each time , the solution x ( t ) corresponds to one point on this curve since if x ( t ) is known so is dx / dt . As time changes , the point ...
... dx / dt vs. x space , for each value of E : dx dt 293 X Figure 20-1 Typical energy curve . For each time , the solution x ( t ) corresponds to one point on this curve since if x ( t ) is known so is dx / dt . As time changes , the point ...
Page 98
... dx + c + kx dt 2 dt = 0 . ( a ) Show that E = m / 2 ( dx / dt ) 2 + ( k / 2 ) x2 is a decreasing function of time . ( b ) Let v = dx / dt and show that dʊ / dx = ( -cv - - kx ) / mv . ( c ) Show that if c2 < 4mk , then v = Ax is not a ...
... dx + c + kx dt 2 dt = 0 . ( a ) Show that E = m / 2 ( dx / dt ) 2 + ( k / 2 ) x2 is a decreasing function of time . ( b ) Let v = dx / dt and show that dʊ / dx = ( -cv - - kx ) / mv . ( c ) Show that if c2 < 4mk , then v = Ax is not a ...
Contents
NEWTONS | 6 |
OSCILLATION OF A SPRINGMASS SYSTEM | 12 |
QUALITATIVE AND QUANTITATIVE BEHAVIOR | 18 |
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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amplitude of oscillation analyze approximation Assume birth c₁ calculation characteristics Consider constant coefficient corresponding d2x dt2 damping de/dt decreases delay depend derived determine difference equation discussed dx dt dx/dt energy integral equilibrium population equilibrium solution equivalent example exercise exponential Figure formula function growth rate Hint increases initial conditions initial value problem isoclines linearized stability analysis logistic equation mass mathematical model maximum method of characteristics motion moving N₁ Newton's nonlinear pendulum number of cars obtained occur ordinary differential equations oscillation P₁ partial differential equation period phase plane Pmax population growth potential energy r₁ result sharks shock Show shown in Fig simple harmonic motion Sketch the solution sketched in Fig slope solution curves solve species spring spring-mass system stable straight line Suppose Taylor series tion traffic flow trajectories Umax unstable equilibrium position variables vector x₁ yields zero