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Math Advanced Math Fundamentals of Differential Equations (9th Edition)

(a) To show: The Wronskian W [ y 1 , y 2 ] can be written as the 2 × 2 determinant W [ y 1 , y 2 ] ( t ) = | y 1 ( t ) y 2 ( t ) y 1 ′ ( t ) y 2 ′ ( t ) | . Let y 1 ( t ) and y 2 ( t ) be two differentiable functions. Then W [ y 1 , y 2 ] ( t ) = y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) . Note that, by determinant of a 2 × 2 matrix [ a b c d ] is defined by | a b c d | = a d − c b . Thus, the value of | y 1 ( t ) y 2 ( t ) y 1 ′ ( t ) y 2 ′ ( t ) | will be y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) , which is the Wronskian of y 1 ( t ) and y 2 ( t ) . Therefore, W [ y 1 , y 2 ] ( t ) = | y 1 ( t ) y 2 ( t ) y 1 ′ ( t ) y 2 ′ ( t ) | . (b) To prove: The functions y 1 ( t ) and y 2 ( t ) are linearly independent on I if and only if their Wronskian is never zero on I . Definition used: A pair of functions y 1 ( t ) and y 2 ( t ) is said to be linearly independent on the interval I if and only if neither of them is a constant multiple of the other on all of I . Result used: For any real numbers a ( ≠ 0 ) , b and c , if y 1 ( t ) and y 2 ( t ) are any two solutions to the differential equation a y ″ + b y ′ + c y = 0 on ( − ∞ , ∞ ) and if the equality y 1 ( τ ) y 2 ′ ( τ ) − y 1 ′ ( τ ) y 2 ( τ ) = 0 holds at any point τ , then y 1 and y 2 are linearly dependent on ( − ∞ , ∞ ) . Description: Suppose that y 1 ( t ) and y 2 ( t ) are linearly independent. By the above mentioned definition, the functions y 1 ( t ) and y 2 ( t ) cannot be expressed as constant multiple of each other. That is, there does not exist k such that y 1 ( t ) = k y 2 ( t ) . In other words, y 1 ( t ) ≠ k y 2 ( t ) . Since they are not constant multiples of each other, the Wronskian W [ y 1 , y 2 ] ( t ) = y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) will not be equal to zero. Otherwise, it will contradicts the above mentioned result. Suppose that the Wronskian is not zero. That is, y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) ≠ 0 y 1 ( t ) y 2 ′ ( t ) ≠ y 1 ′ ( t ) y 2 ( t ) Thus, y 2 ( t ) is not a constant multiple of y 1 ( t ) and vice versa. That is, y 1 ( t ) and y 2 ( t ) are linearly independent. Therefore, the functions y 1 ( t ) and y 2 ( t ) are linearly independent on I if and only if their Wronskian is never zero on I . (c) To show: If y 1 ( t ) and y 2 ( t ) are any two differentiable functions that are linearly dependent on I , then their Wronskian is identically zero on I . Suppose that y 1 ( t ) and y 2 ( t ) are linearly dependent. By the above mentioned definition, the functions y 1 ( t ) and y 2 ( t ) can be expressed as constant multiple of each other. That is, there exist k such that y 1 ( t ) = k y 2 ( t ) . On differentiation, y 1 ′ ( t ) = k y 2 ′ ( t ) . Then, the Wronskian W [ y 1 , y 2 ] ( t ) = y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) will be, y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) = k y 2 ( t ) y 2 ′ ( t ) − k y 2 ′ ( t ) y 2 ( t ) ( ∵ y 1 ( t ) = k y 2 ( t ) and y 1 ′ ( t ) = k y 2 ′ ( t ) ) = k y 2 ( t ) y 2 ′ ( t ) − k y 2 ( t ) y 2 ′ ( t ) = 0 Therefore, if y 1 ( t ) and y 2 ( t ) are any two differentiable functions that are linearly dependent on I , then their Wronskian is identically zero on I .

(a) To show: The Wronskian W [ y 1 , y 2 ] can be written as the 2 × 2 determinant W [ y 1 , y 2 ] ( t ) = | y 1 ( t ) y 2 ( t ) y 1 ′ ( t ) y 2 ′ ( t ) | . Let y 1 ( t ) and y 2 ( t ) be two differentiable functions. Then W [ y 1 , y 2 ] ( t ) = y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) . Note that, by determinant of a 2 × 2 matrix [ a b c d ] is defined by | a b c d | = a d − c b . Thus, the value of | y 1 ( t ) y 2 ( t ) y 1 ′ ( t ) y 2 ′ ( t ) | will be y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) , which is the Wronskian of y 1 ( t ) and y 2 ( t ) . Therefore, W [ y 1 , y 2 ] ( t ) = | y 1 ( t ) y 2 ( t ) y 1 ′ ( t ) y 2 ′ ( t ) | . (b) To prove: The functions y 1 ( t ) and y 2 ( t ) are linearly independent on I if and only if their Wronskian is never zero on I . Definition used: A pair of functions y 1 ( t ) and y 2 ( t ) is said to be linearly independent on the interval I if and only if neither of them is a constant multiple of the other on all of I . Result used: For any real numbers a ( ≠ 0 ) , b and c , if y 1 ( t ) and y 2 ( t ) are any two solutions to the differential equation a y ″ + b y ′ + c y = 0 on ( − ∞ , ∞ ) and if the equality y 1 ( τ ) y 2 ′ ( τ ) − y 1 ′ ( τ ) y 2 ( τ ) = 0 holds at any point τ , then y 1 and y 2 are linearly dependent on ( − ∞ , ∞ ) . Description: Suppose that y 1 ( t ) and y 2 ( t ) are linearly independent. By the above mentioned definition, the functions y 1 ( t ) and y 2 ( t ) cannot be expressed as constant multiple of each other. That is, there does not exist k such that y 1 ( t ) = k y 2 ( t ) . In other words, y 1 ( t ) ≠ k y 2 ( t ) . Since they are not constant multiples of each other, the Wronskian W [ y 1 , y 2 ] ( t ) = y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) will not be equal to zero. Otherwise, it will contradicts the above mentioned result. Suppose that the Wronskian is not zero. That is, y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) ≠ 0 y 1 ( t ) y 2 ′ ( t ) ≠ y 1 ′ ( t ) y 2 ( t ) Thus, y 2 ( t ) is not a constant multiple of y 1 ( t ) and vice versa. That is, y 1 ( t ) and y 2 ( t ) are linearly independent. Therefore, the functions y 1 ( t ) and y 2 ( t ) are linearly independent on I if and only if their Wronskian is never zero on I . (c) To show: If y 1 ( t ) and y 2 ( t ) are any two differentiable functions that are linearly dependent on I , then their Wronskian is identically zero on I . Suppose that y 1 ( t ) and y 2 ( t ) are linearly dependent. By the above mentioned definition, the functions y 1 ( t ) and y 2 ( t ) can be expressed as constant multiple of each other. That is, there exist k such that y 1 ( t ) = k y 2 ( t ) . On differentiation, y 1 ′ ( t ) = k y 2 ′ ( t ) . Then, the Wronskian W [ y 1 , y 2 ] ( t ) = y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) will be, y 1 ( t ) y 2 ′ ( t ) − y 1 ′ ( t ) y 2 ( t ) = k y 2 ( t ) y 2 ′ ( t ) − k y 2 ′ ( t ) y 2 ( t ) ( ∵ y 1 ( t ) = k y 2 ( t ) and y 1 ′ ( t ) = k y 2 ′ ( t ) ) = k y 2 ( t ) y 2 ′ ( t ) − k y 2 ( t ) y 2 ′ ( t ) = 0 Therefore, if y 1 ( t ) and y 2 ( t ) are any two differentiable functions that are linearly dependent on I , then their Wronskian is identically zero on I .

Solution Summary: The author illustrates how the Wronskian Wleft[y_1,Y2right] can be written as the 2times 2 determinant.

Fundamentals of Differential Equations (9th Edition)

Fundamentals of Differential Equations (9th Edition)

9th Edition

ISBN: 9780321977069

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Publisher: PEARSON

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Chapter 4.2, Problem 33E

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Chapter 4 Solutions

Fundamentals of Differential Equations (9th Edition)

Ch. 4.1 - Verify that for b = 0 and Fext(t) = 0, equation...Ch. 4.1 - If Fext(t) = 0, equation (3) becomes my + by + ky...Ch. 4.1 - Show that if Fext(t) = 0, m = 1, k = 9, and b = 6,...Ch. 4.1 - Verify that y = sin 3t + 2 cos 3t is a solution to...Ch. 4.1 - Prob. 5E Ch. 4.1 - An external force F(t) = 2 cos 2t is applied to a...Ch. 4.1 - Prob. 7E Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - Prob. 10E

Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - Prob. 3E Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - Prob. 5E Ch. 4.2 - Prob. 6E Ch. 4.2 - Prob. 7E Ch. 4.2 - Prob. 8E Ch. 4.2 - Prob. 9E Ch. 4.2 - Prob. 10E Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 15E Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 17E Ch. 4.2 - Prob. 18E Ch. 4.2 - Prob. 19E Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 21E Ch. 4.2 - Prob. 22E Ch. 4.2 - Prob. 23E Ch. 4.2 - Prob. 24E Ch. 4.2 - Prob. 25E Ch. 4.2 - Prob. 26E Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - Prob. 31E Ch. 4.2 - Prob. 32E Ch. 4.2 - Prob. 33E Ch. 4.2 - Prob. 34E Ch. 4.2 - Prob. 35E Ch. 4.2 - Using the definition in Problem 35, prove that if...Ch. 4.2 - Prob. 37E Ch. 4.2 - Prob. 38E Ch. 4.2 - Prob. 39E Ch. 4.2 - Prob. 40E Ch. 4.2 - In Problems 3741, find three linearly independent...Ch. 4.2 - Prob. 42E Ch. 4.2 - Prob. 43E Ch. 4.2 - Solve the initial value problem: y 2y y + 2y =...Ch. 4.2 - Prob. 45E Ch. 4.2 - Prob. 46E Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - Prob. 5E Ch. 4.3 - Prob. 6E Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 920, find a general solution. 9. y 8y...Ch. 4.3 - In Problems 920, find a general solution. 10. y +...Ch. 4.3 - Prob. 11E Ch. 4.3 - Prob. 12E Ch. 4.3 - In Problems 920, find a general solution. 13. y ...Ch. 4.3 - Prob. 14E Ch. 4.3 - Prob. 15E Ch. 4.3 - Prob. 16E Ch. 4.3 - In Problems 920, find a general solution. 17. y y...Ch. 4.3 - In Problems 920, find a general solution. 18. 2y +...Ch. 4.3 - Prob. 19E Ch. 4.3 - In Problems 920, find a general solution. 20. y y...Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 22E Ch. 4.3 - Prob. 23E Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 25E Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 27E Ch. 4.3 - Prob. 28E Ch. 4.3 - Prob. 29E Ch. 4.3 - Prob. 30E Ch. 4.3 - Prob. 31E Ch. 4.3 - Prob. 32E Ch. 4.3 - Prob. 33E Ch. 4.3 - Prob. 34E Ch. 4.3 - Swinging Door. The motion of a swinging door with...Ch. 4.3 - Prob. 36E Ch. 4.3 - Prob. 37E Ch. 4.3 - Prove the sum of angles formula for the sine...Ch. 4.4 - In Problems 18, decide whether or not the method...Ch. 4.4 - Prob. 2E Ch. 4.4 - Prob. 3E Ch. 4.4 - Prob. 4E Ch. 4.4 - Prob. 5E Ch. 4.4 - Prob. 6E Ch. 4.4 - Prob. 7E Ch. 4.4 - Prob. 8E Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - Prob. 16E Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - Prob. 18E Ch. 4.4 - Prob. 19E Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - In Problems 926, find a particular solution to the...Ch. 4.4 - Prob. 23E Ch. 4.4 - Prob. 24E Ch. 4.4 - Prob. 25E Ch. 4.4 - Prob. 26E Ch. 4.4 - Prob. 27E Ch. 4.4 - Prob. 28E Ch. 4.4 - Prob. 29E Ch. 4.4 - Prob. 30E Ch. 4.4 - Prob. 31E Ch. 4.4 - Prob. 32E Ch. 4.4 - Prob. 33E Ch. 4.4 - Prob. 34E Ch. 4.4 - Prob. 35E Ch. 4.4 - In Problems 3336, use the method of undetermined...Ch. 4.5 - Given that y1(t) = cos t is a solution to y y + y...Ch. 4.5 - Prob. 2E Ch. 4.5 - Prob. 3E Ch. 4.5 - Prob. 4E Ch. 4.5 - Prob. 5E Ch. 4.5 - Prob. 6E Ch. 4.5 - Prob. 7E Ch. 4.5 - In Problems 38, a nonhomogeneous equation and a...Ch. 4.5 - Prob. 9E Ch. 4.5 - Prob. 10E Ch. 4.5 - Prob. 11E Ch. 4.5 - Prob. 12E Ch. 4.5 - Prob. 13E Ch. 4.5 - In Problems 916 decide whether the method of...Ch. 4.5 - Prob. 15E Ch. 4.5 - Prob. 16E Ch. 4.5 - Prob. 17E Ch. 4.5 - Prob. 18E Ch. 4.5 - Prob. 19E Ch. 4.5 - In Problems 1722, find a general solution to the...Ch. 4.5 - Prob. 21E Ch. 4.5 - Prob. 22E Ch. 4.5 - Prob. 23E Ch. 4.5 - Prob. 24E Ch. 4.5 - Prob. 25E Ch. 4.5 - Prob. 26E Ch. 4.5 - Prob. 27E Ch. 4.5 - Prob. 28E Ch. 4.5 - Prob. 29E Ch. 4.5 - Prob. 30E Ch. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - Prob. 33E Ch. 4.5 - In Problems 3136, determine the form of a...Ch. 4.5 - Prob. 35E Ch. 4.5 - Prob. 36E Ch. 4.5 - Prob. 37E Ch. 4.5 - Prob. 38E Ch. 4.5 - In Problems 3740, find a particular solution to...Ch. 4.5 - Prob. 40E Ch. 4.5 - Prob. 41E Ch. 4.5 - Prob. 42E Ch. 4.5 - Prob. 43E Ch. 4.5 - Prob. 44E Ch. 4.5 - Prob. 46E Ch. 4.5 - Prob. 47E Ch. 4.5 - Prob. 48E Ch. 4.6 - Prob. 1E Ch. 4.6 - In Exercises 12, rectangles have been drawn to...Ch. 4.6 - Prob. 3E Ch. 4.6 - Prob. 4E Ch. 4.6 - Prob. 5E Ch. 4.6 - Prob. 6E Ch. 4.6 - Prob. 7E Ch. 4.6 - Prob. 8E Ch. 4.6 - Prob. 9E Ch. 4.6 - Prob. 10E Ch. 4.6 - Prob. 11E Ch. 4.6 - Prob. 12E Ch. 4.6 - Prob. 13E Ch. 4.6 - Prob. 14E Ch. 4.6 - Prob. 15E Ch. 4.6 - In Exercises 1318, use a calculator or a computer...Ch. 4.6 - Prob. 17E Ch. 4.6 - In Exercises 1318, use a calculator or a computer...Ch. 4.6 - For f(x) = 3x + 2, evaluate the Riemann sums: (a)...Ch. 4.6 - For g(x) = 4x 1, evaluate the Riemann sums: (a)...Ch. 4.6 - Prob. 21E Ch. 4.6 - Use Table 5.9 to evaluate the Riemann sums: Table...Ch. 4.6 - The graph of f(t) is in Figure 5.40. Which of the...Ch. 4.6 - Prob. 24E Ch. 4.6 - Prob. 25E Ch. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - Prob. 3E Ch. 4.7 - In Problems 1 through 4, use Theorem 5 to discuss...Ch. 4.7 - Prob. 5E Ch. 4.7 - In Problems 5 through 8, determine whether Theorem...Ch. 4.7 - Prob. 7E Ch. 4.7 - In Problems 5 through 8, determine whether Theorem...Ch. 4.7 - Prob. 9E Ch. 4.7 - In Problems 9 through 14, find a general solution...Ch. 4.7 - Prob. 11E Ch. 4.7 - In Problems 9 through 14, find a general solution...Ch. 4.7 - Prob. 13E Ch. 4.7 - In Problems 9 through 14, find a general solution...Ch. 4.7 - In Problems 15 through 18, find a general solution...Ch. 4.7 - In Problems 15 through 18, find a general solution...Ch. 4.7 - Prob. 17E Ch. 4.7 - Prob. 18E Ch. 4.7 - Prob. 19E Ch. 4.7 - Prob. 20E Ch. 4.7 - Prob. 21E Ch. 4.7 - Prob. 22E Ch. 4.7 - Prob. 23E Ch. 4.7 - Solve the following CauchyEuler equations by using...Ch. 4.7 - Prob. 25E Ch. 4.7 - Prob. 26E Ch. 4.7 - Prob. 27E Ch. 4.7 - Prob. 28E Ch. 4.7 - Prove that if y1 and y2 are linearly independent...Ch. 4.7 - Prob. 30E Ch. 4.7 - Prob. 31E Ch. 4.7 - Prob. 32E Ch. 4.7 - Prob. 33E Ch. 4.7 - Prob. 34E Ch. 4.7 - Prob. 35E Ch. 4.7 - Prob. 36E Ch. 4.7 - Prob. 37E Ch. 4.7 - Prob. 38E Ch. 4.7 - Prob. 39E Ch. 4.7 - Prob. 40E Ch. 4.7 - Prob. 41E Ch. 4.7 - In Problems 41 through 44, a differential equation...Ch. 4.7 - Prob. 43E Ch. 4.7 - Prob. 44E Ch. 4.7 - Prob. 45E Ch. 4.7 - Prob. 46E Ch. 4.7 - Prob. 47E Ch. 4.7 - Prob. 48E Ch. 4.7 - Prob. 49E Ch. 4.7 - Prob. 50E Ch. 4.7 - Prob. 51E Ch. 4.7 - Prob. 52E Ch. 4.8 - Prob. 1E Ch. 4.8 - Prob. 2E Ch. 4.8 - Try to predict the qualitative features of the...Ch. 4.8 - Prob. 4E Ch. 4.8 - Prob. 5E Ch. 4.8 - Prob. 6E Ch. 4.8 - Prob. 7E Ch. 4.8 - Prob. 8E Ch. 4.8 - Prob. 9E Ch. 4.8 - Prob. 10E Ch. 4.8 - Prob. 11E Ch. 4.8 - Use reduction of order to derive the solution...Ch. 4.8 - Prob. 13E Ch. 4.8 - Prob. 14E Ch. 4.8 - Prob. 15E Ch. 4.8 - Prob. 16E Ch. 4.8 - Prob. 17E Ch. 4.9 - Prob. 1E Ch. 4.9 - All problems refer to the massspring configuration...Ch. 4.9 - Prob. 3E Ch. 4.9 - Prob. 4E Ch. 4.9 - Prob. 5E Ch. 4.9 - Prob. 6E Ch. 4.9 - Prob. 7E Ch. 4.9 - Prob. 8E Ch. 4.9 - All problems refer to the massspring configuration...Ch. 4.9 - Prob. 10E Ch. 4.9 - Prob. 11E Ch. 4.9 - Prob. 12E Ch. 4.9 - Prob. 13E Ch. 4.9 - Prob. 14E Ch. 4.9 - Prob. 15E Ch. 4.9 - Prob. 16E Ch. 4.9 - Prob. 17E Ch. 4.9 - Prob. 18E Ch. 4.10 - In the following problems, take g = 32 ft/sec2 for...Ch. 4.10 - Prob. 2E Ch. 4.10 - Prob. 3E Ch. 4.10 - Prob. 4E Ch. 4.10 - Prob. 5E Ch. 4.10 - Prob. 6E Ch. 4.10 - Prob. 7E Ch. 4.10 - Prob. 8E Ch. 4.10 - In the following problems, take g = 32 ft/sec2 for...Ch. 4.10 - Prob. 10E Ch. 4.10 - Prob. 11E Ch. 4.10 - Prob. 12E Ch. 4.10 - In the following problems, take g = 32 ft/sec2 for...Ch. 4.10 - Prob. 14E Ch. 4.10 - Prob. 15E Ch. 4 - In Problems 128, find a general solution to the...Ch. 4 - Prob. 2RP Ch. 4 - Prob. 3RP Ch. 4 - Prob. 4RP Ch. 4 - Prob. 5RP Ch. 4 - Prob. 6RP Ch. 4 - Prob. 7RP Ch. 4 - Prob. 8RP Ch. 4 - Prob. 9RP Ch. 4 - Prob. 10RP Ch. 4 - Prob. 11RP Ch. 4 - Prob. 12RP Ch. 4 - Prob. 13RP Ch. 4 - Prob. 14RP Ch. 4 - Prob. 15RP Ch. 4 - Prob. 16RP Ch. 4 - Prob. 17RP Ch. 4 - In Problems 128, find a general solution to the...Ch. 4 - Prob. 19RP Ch. 4 - Prob. 20RP Ch. 4 - Prob. 21RP Ch. 4 - Prob. 22RP Ch. 4 - Prob. 23RP Ch. 4 - Prob. 24RP Ch. 4 - Prob. 25RP Ch. 4 - Prob. 26RP Ch. 4 - Prob. 27RP Ch. 4 - Prob. 28RP Ch. 4 - Prob. 29RP Ch. 4 - Prob. 30RP Ch. 4 - Prob. 31RP Ch. 4 - Prob. 32RP Ch. 4 - Prob. 33RP Ch. 4 - Prob. 34RP Ch. 4 - Prob. 35RP Ch. 4 - Prob. 36RP Ch. 4 - Prob. 37RP Ch. 4 - Prob. 38RP Ch. 4 - A 32-lb weight is attached to a vertical spring,...Ch. 4 - Prob. 1TWE Ch. 4 - Prob. 2TWE Ch. 4 - Prob. 3TWE Ch. 4 - For students with a background in linear algebra:...

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