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Get Full Access to Elementary Differential Equations And Boundary Value Problems - 11 Edition - Chapter 7.1 - Problem 16
Get Full Access to Elementary Differential Equations And Boundary Value Problems - 11 Edition - Chapter 7.1 - Problem 16

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# Consider the circuit shown in Figure 7.1.2. Let I1, I2, and I3 be the currents through ISBN: 9781119256007 392

## Solution for problem 16 Chapter 7.1

Elementary Differential Equations and Boundary Value Problems | 11th Edition

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Problem 16

Consider the circuit shown in Figure 7.1.2. Let I1, I2, and I3 be the currents through the capacitor, resistor, and inductor, respectively. Likewise, let V1, V2, and V3 be the corresponding voltage drops. The arrows denote the arbitrarily chosen directions in which currents and voltage drops will be taken to be positive. a. Applying Kirchhoffs second law to the upper loop in the circuit, show that V1 V2 = 0. (15) In a similar way, show that V2 V3 = 0. (16) b. Applying Kirchhoffs first law to either node in the circuit, show that I1 + I2 + I3 = 0. (17) c. Use the current-voltage relation through each element in the circuit to obtain the equations CV_ 1 = I1, V2 = RI2, L I_ 3 = V3. (18) d. Eliminate V2, V3, I1, and I2 among equations (15) through (18) to obtain CV_ 1 = I3 V1 R , L I_ 3 = V1. (19) Observe that if we omit the subscripts in equations (19), then we have the system (2) of this section. 1

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Section 1.5 Reduced Echelon Form 10 r If we get   then the solution is 01 s x1= r and x =2s.  0 r  0 s If we get 001 t then the solution is   x1= r, x2= s and x =3t. With many systems, particularly rectangular systems, this isn't possible. We need a form to show us when the augmented matrix is "as reduced as possible". Reduced Echelon Form All four of these conditions must be met: (1) All rows that consist entirely of zeros are at the bottom. (2) In a nonzero row, the first nonzero entry from the left is a 1. This is called a "leading one". (3) The leading ones form a stairstep pattern down and right. (4) The leading one in each

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