1550078481-Ordinary_Differential_Equations__Roberts_

Chapter 9 Applications of Linear Systems with Constant Coefficients In this chapter we present several applications which requir ...

384 Ordinary Differential Equations For i = 1, 2 let Yi represent the vertical displacement of mass mi from its equilibrium posi ...

Applications of Linear Systems with Constant Coefficients 385 SOLUTION For the given values of m 1 , m2, k 1 , and k 2 system (3 ...

386 Ordinary Differential Equations where /3 = 2.66316, "( =. 712450 , and ( 0 .3 75485 ) 0.000000 -0.0 347007 ) 0.000000 ( 0.21 ...

Applications of Linear Systems with Constant Coefficients 387 where {3 = 2.66316 and "( = .712450. A graph of the position of th ...

388 Ordinary Differential Equations Exercise 1. Write the real general solution for the coupled spring-mass system (3) for a. m ...

Applications of Linear Systems with Constant Coefficients 389 System at Equilibrium System at Timet .. I I Figure 9.4 A Coupled ...

390 Ordinary Differential Equations 9.2 Pendulum Systems Couple d P endulums A coupled pendulum system consists of a pair of ide ...

Applications of Linear Systems with Constant Coefficients 391 0 1 0 0 u' 1 U1 (~ + ~)^0 k 0 u~ m U2 (9) u' 3 0 0 0 1 U3 u' k - ...

392 Ordinary Differential Equations Exercise 7. a. Write system (10) as an equivalent system of first-order dif- ferential equat ...

Applications of Linear Systems with Constant Coefficients 393 EXAMPLE 4 Solving the Nonhomogeneous Initial Value Problem (12) Fo ...

394 Ordinary Differential Equations nonhomogeneous differential equations with constant coefficients. Recall that the method con ...

Applications of Linear Systems with Constant Coefficients 395 Substituting for Up and u~ in (13a), u~ = Aup + b , results in or ...

396 Ordinary Differential Equations Or Thus, c 1 , c2, c3 and c 4 must simultaneously satisfy C1 + C4/2 = 0 -C3 = 0 C2 + C3/2 = ...

Applications of Linear Systems with Constant Coefficients 397 5 y 4 3 2 1 2 0. 4 0. 6 0. 8 1 1. 2 1. 4 F igure 9.7 The Path of ...

398 Ordinary Differential Equations the following assumptions: The rate of change of the amount of substance in any container at ...

Applications of Linear Systems with Constant Coefficients 399 which the solution enters the container from the other container m ...

400 Ordinary Differential Equations assume both containers are always full. This means the sum of the input rates to a container ...

Applications of Linear Systems with Constant Coefficients 401 Exercise 9. a. Find the general solution to the homogeneo us syste ...

402 Ordinary Differential Equations Exercise 13. Three small ponds each containing 10 , 000 gallons of pure water are formed by ...