Differential Equations
4th Edition
ISBN: 9780495561989
Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall
Publisher: Cengage Learning
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Chapter 3.2, Problem 15E
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Use mathematical induction to prove the following statement: For all natural numbers n, 5 divides 6^n - 1 (show every step in detail)
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Chapter 3 Solutions
Differential Equations
Ch. 3.1 - Recall the model dx dt=ax+by dy dt=cx+dy for...Ch. 3.1 - In Exercises 57 , rewrite the specified linear...Ch. 3.1 - In Exercises 57 , rewrite the specified linear...Ch. 3.1 - In Exercises 57 , rewrite the specified linear...Ch. 3.1 - In Exercises 89 , rewrite the specified linear...Ch. 3.1 - For the linear systems given in Exercises 1013,...Ch. 3.1 - For the linear systems given in Exercises 1013,...Ch. 3.1 - Prob. 13ECh. 3.1 - Let A=(abcd) be a nonzero matrix. That is, suppose...Ch. 3.1 - The general form of a linear, homogeneous,...
Ch. 3.1 - Convert the third-order differential equation $...Ch. 3.1 - Consider the linear system dYdt=(2011)Y Show that...Ch. 3.1 - Consider the linear system dYdt=(1 113)Y (a)Show...Ch. 3.1 - A=( 2 33 2) Functions: Y1(t)=e2t(cos3t,sin3t)...Ch. 3.2 - In Exercises 110 (a) compute the eigenvalues; (b)...Ch. 3.2 - In Exercises 110 (a) compute the eigenvalues; (b)...Ch. 3.2 - In Exercises 110 (a) compute the eigenvalues; (b)...Ch. 3.2 - In Exercises 110 (a) compute the eigenvalues; (b)...Ch. 3.2 - In Exercises 110 (a) compute the eigenvalues; (b)...Ch. 3.2 - In Exercises 110 (a) compute the eigenvalues; (b)...Ch. 3.2 - In Exercises 110 (a) compute the eigenvalues; (b)...Ch. 3.2 - In Exercises $1-10$ (a) compute the eigenvalues;...Ch. 3.2 - Solve the initial-value problem dx dt=2x2y dy...Ch. 3.2 - Solve the initial-value problem dYdt=( 412...Ch. 3.2 - Show that a is the only eigenvalue and that every...Ch. 3.2 - A matrix of the form A=(ab0d) is called upper...Ch. 3.2 - A matrix of the form B=(abbd) is called symmetric....Ch. 3.2 - Consider the second-order equation...Ch. 3.2 - For the harmonic oscillator with mass m=1, spring...Ch. 3.2 - In Exercises 21-24, we return to Exercises 1-4 in...Ch. 3.3 - In Exercises 18, we refer to linear systems from...Ch. 3.3 - In Exercises 18, we refer to linear systems from...Ch. 3.3 - In Exercises 18, we refer to linear systems from...Ch. 3.3 - In Exercises 1-8, we refer to linear systems from...Ch. 3.3 - In Exercises 912, we refer to initial-value...Ch. 3.3 - In Exercises 13-16, we refer to the second-order...Ch. 3.3 - The slope field for the system dx dt=2x+12y dy...Ch. 3.3 - Consider the linear system dYdt=( 2102)Y $ (a)...Ch. 3.4 - Suppose that the 22 matrix A has =1+3i as an...Ch. 3.4 - Suppose that the 22 matrix B has =2+5i as an...Ch. 3.4 - In Exercises 3-8, each linear system has complex...Ch. 3.4 - In Exercises 3-8, each linear system has complex...Ch. 3.4 - In Exercises 3-8, each linear system has complex...Ch. 3.4 - In Exercises 3-8, each linear system has complex...Ch. 3.4 - In Exercises 3-8, each linear system has complex...Ch. 3.4 - In Exercises 9-14, the linear systems are the same...Ch. 3.4 - In Exercises 9-14, the linear systems are the same...Ch. 3.4 - In Exercises 9-14, the linear systems are the same...Ch. 3.5 - In Exercises 1-4, each of the linear systems has...Ch. 3.5 - In Exercises 5-8, the linear systems are the same...Ch. 3.5 - Given a quadratic 2++, what condition on and ...Ch. 3.6 - In Exercises 16, find the general solution (in...Ch. 3.6 - In Exercises 16, find the general solution (in...Ch. 3.6 - In Exercises 16, find the general solution (in...Ch. 3.6 - In Exercises 712, find the solution of the given...Ch. 3.6 - In Exercises 712, find the solution of the given...Ch. 3.6 - In Exercises 712, find the solution of the given...Ch. 3.6 - In Exercises 712 , find the solution of the given...Ch. 3.6 - In Exercises 1320, consider harmonic oscillators...Ch. 3.6 - In Exercises 13-20, consider harmonic oscillators...Ch. 3.6 - In Exercises 1320, consider harmonic oscillators...Ch. 3.7 - In Exercises 27 , we consider the one-parameter...Ch. 3.7 - In Exercises 2-7, we consider the one-parameter...
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