To determine The current from the given figure. Explanation Let the currents in the left and right loops be I 1 ( t ) and I 2 ( t ) . From the figure, the voltage drops in the right loop are L I ′ 2 and R ( I 2 − I 1 ) . Note that, by the sum of the voltage drop over the capacitor is zero. By the note, the equation becomes as L I ′ 2 + R ( I 2 − I 1 ) = 0 . Given that, R = 1 Ω , L = 1.25 H, C = 0.2 F with initial conditions on current as [ I 1 ( 0 ) I 2 ( 0 ) ] = [ 1 1 ] . Substitute those values in the equation L I ′ 2 + R ( I 2 − I 1 ) = 0 and solve for I ′ 1 . ( 1.25 ) I ′ 2 + ( I 2 − I 1 ) = 0 1.25 I ′ 2 = I 1 − I 2 I ′ 2 = 1 1.25 I 1 − 1 1.25 I 2 I ′ 2 = 0.8 I 1 − 0.8 I 2 From the figure, the voltage drops in the left loop are R ( I 1 − I 2 ) and 1 C ∫ I 1 d t . The sum of the voltage drop over the capacitor is zero. That is, R ( I 1 − I 2 ) + 1 C ∫ I 1 d t = 0 . Differentiate the equation R ( I 2 − I 1 ) + 1 C ∫ I 2 d t = 0 both sides with respect to t . d d t ( R ( I 1 − I 2 ) ) + d d t ( 1 C ∫ I 1 d t ) = d d t ( 0 ) R ( I ′ 1 − I ′ 2 ) + 1 C I 1 = 0 R I ′ 1 − R I ′ 2 + I 1 C = 0 Substitute R = 1 and C = 0.2 in the equation R I ′ 1 − R I ′ 2 + I 1 C = 0 . I ′ 1 − I ′ 2 + I 1 0.2 = 0 I ′ 1 = I ′ 2 − 5 I 1 Substitute the value of I ′ 2 = 0.8 I 1 − 0.8 I 2 in the equation I ′ 1 = I ′ 2 − 5 I 1 . I ′ 1 = 0.8 I 1 − 0.8 I 2 − 5 I 1 = − 4.2 I 1 − 0.8 I 2 Now, the system of differential equations in terms currents is { I ′ 1 = − 4.2 I 1 − 0.8 I 2 I ′ 2 = 0.8 I 1 − 0.8 I 2 . Solve the system of differential equation as follows. The coefficient matrix of the corresponding systems of differential equations is A = [ − 4.2 − 0.8 0.8 − 0.8 ] . Hence, a single vector equation is I ′ = [ − 4.2 − 0.8 0.8 − 0.8 ] I , where I = [ I 1 I 2 ] . Characteristic equation of the coefficient matrix A = [ − 4.2 − 0.8 0.8 − 0.8 ] is det ( A − λ I ) = 0 , where λ eigenvalue and I is 2 × 2 identity matrix . Simplify for the det ( A − λ I ) = 0 using the coefficient matrix as follows. det ( A − λ I ) = 0 det ( [ − 4.2 − 0.8 0.8 − 0.8 ] − λ [ 1 0 0 1 ] ) = 0 det ( [ − 4.2 − λ − 0.8 0.8 − 0.8 − λ ] ) = 0 ( λ + 4.2 ) ( λ + 0.8 ) + 0.64 = 0 Further simplified as, λ 2 + 5 λ + 4 = 0 ( λ + 4 ) ( λ + 1 ) = 0 λ + 4 = 0 or λ + 1 = 0 λ = − 4 or λ = − 1 Compute the eigenvector t ( 1 ) = [ t 1 t 2 ] corresponding to the eigenvalue λ = − 1 as follows

10th Edition

ISBN: 9781119809210

Author: Kreyszig

Publisher: WILEY

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3. Consider the following theorem: Theorem: If n is an odd integer, then n³ is an odd integer. Note: There is an implicit universal quantifier for this theorem. Technically we could write: For all integers n, if n is an odd integer, then n³ is an odd integer. (a) Explore the statement by constructing at least three examples that satisfy the hypothesis, one of which uses a negative value. Verify the conclusion is true for each example. You do not need to write your examples formally, but your work should be easy to follow. (b) Pick one of your examples from part (a) and complete the following sentence frame: One example that verifies the theorem is when n = We see the hypothesis is true because and the conclusion is true because (c) Use the definition of odd to construct a know-show table that outlines the proof of the theorem. You do not need to write a proof at this time.

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ADV.ENG.MATH (LL) W/WILEYPLUS BUNDLE

Ch. 4.1 - Prob. 1P Ch. 4.1 - Prob. 2P Ch. 4.1 - Prob. 3P Ch. 4.1 - Prob. 4P Ch. 4.1 - If you extend Example 1 by a tank T3 of the same...Ch. 4.1 - Find a “general solution” of the system in Prob....Ch. 4.1 - In Example 2 find the currents: 7. If the initial...Ch. 4.1 - Prob. 8P Ch. 4.1 - Prob. 9P Ch. 4.1 - Find a general solution of the given ODE (a) by...

Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Prob. 14P Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - Find a real general solution of the following...Ch. 4.3 - Prob. 9P Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - 10–15 IVPs Solve the following initial value...Ch. 4.3 - Prob. 12P Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - Prob. 16P Ch. 4.3 - Prob. 17P Ch. 4.3 - Prob. 18P Ch. 4.3 - Prob. 19P Ch. 4.4 - Prob. 1P Ch. 4.4 - Prob. 2P Ch. 4.4 - Prob. 3P Ch. 4.4 - Prob. 4P Ch. 4.4 - Prob. 5P Ch. 4.4 - Prob. 6P Ch. 4.4 - Prob. 7P Ch. 4.4 - Prob. 8P Ch. 4.4 - Prob. 9P Ch. 4.4 - Prob. 10P Ch. 4.4 - Prob. 11P Ch. 4.4 - Prob. 12P Ch. 4.4 - Prob. 13P Ch. 4.4 - Prob. 14P Ch. 4.4 - Prob. 15P Ch. 4.4 - Prob. 16P Ch. 4.4 - Prob. 17P Ch. 4.5 - Prob. 1P Ch. 4.5 - Prob. 2P Ch. 4.5 - Prob. 4P Ch. 4.5 - Prob. 5P Ch. 4.5 - Prob. 6P Ch. 4.5 - Prob. 7P Ch. 4.5 - Prob. 8P Ch. 4.5 - Prob. 9P Ch. 4.5 - Prob. 10P Ch. 4.5 - Prob. 11P Ch. 4.5 - Prob. 12P Ch. 4.5 - Prob. 13P Ch. 4.6 - Prob. 1P Ch. 4.6 - Prob. 2P Ch. 4.6 - Prob. 3P Ch. 4.6 - Prob. 4P Ch. 4.6 - Prob. 5P Ch. 4.6 - Prob. 6P Ch. 4.6 - Prob. 7P Ch. 4.6 - Prob. 9P Ch. 4.6 - Prob. 10P Ch. 4.6 - Prob. 11P Ch. 4.6 - Prob. 12P Ch. 4.6 - Prob. 13P Ch. 4.6 - Prob. 14P Ch. 4.6 - Prob. 15P Ch. 4.6 - Prob. 16P Ch. 4.6 - Prob. 17P Ch. 4.6 - Prob. 19P Ch. 4 - Prob. 1RQ Ch. 4 - Prob. 2RQ Ch. 4 - How can you transform an ODE into a system of...Ch. 4 - What are qualitative methods for systems? Why are...Ch. 4 - Prob. 5RQ Ch. 4 - Prob. 6RQ Ch. 4 - What are eigenvalues? What role did they play in...Ch. 4 - Prob. 8RQ Ch. 4 - Prob. 9RQ Ch. 4 - Prob. 10RQ Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Prob. 15RQ Ch. 4 - Prob. 16RQ Ch. 4 - Prob. 17RQ Ch. 4 - Prob. 18RQ Ch. 4 - Prob. 19RQ Ch. 4 - Prob. 20RQ Ch. 4 - Prob. 21RQ Ch. 4 - Prob. 22RQ Ch. 4 - Prob. 23RQ Ch. 4 - Prob. 24RQ Ch. 4 - Prob. 25RQ Ch. 4 - Network. Find the currents in Fig. 103 when R = 1...Ch. 4 - Prob. 27RQ Ch. 4 - Prob. 28RQ Ch. 4 - Find the location and kind of all critical points...Ch. 4 - Find the location and kind of all critical points...

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