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

The displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 , where the system starts from rest and sketch the displacement y ( t ) . The displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 is y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 _ . Procedure used: Method of undetermined coefficients: A particular solution to the differential equation a y ″ + b y ′ + c y = C t m e r t , where m is a nonnegative integer, is of the form y p ( t ) = t s ( A m t m + ⋯ + A 1 t + A 0 ) e r t with: (i) s = 0 , if r is not a root of the associated auxiliary equation; (ii) s = 1 , if r is a simple root of the associated auxiliary equation; (iii) s = 2 , if r is a double root of the associated auxiliary equation. Calculation: Consider the system governed by the differential equation m d 2 y d t 2 + b d y d t + k y = F 0 cos γ t , where F 0 and γ are non-negative constants and b 2 > 4 m k . A solution to the above system has the form y = y h + y p , where y p is a particular solution and y h is a general solution to the corresponding homogeneous equation. Given that, m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 . Substitute m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 in m d 2 y d t 2 + b d y d t + k y = F 0 cos γ t , 2 d 2 y d t 2 + 8 d y d t + 6 y = 18 cos ( 0 × t ) 2 d 2 y d t 2 + 8 d y d t + 6 y = 18 cos ( 0 ) 2 d 2 y d t 2 + 8 d y d t + 6 y = 18 d 2 y d t 2 + 4 d y d t + 3 y = 9 ( divide by 2 on both sides ) That is, y ″ + 4 y ′ + 3 y = 9 . The corresponding homogenous equation is, y ″ + 4 y ′ + 3 y = 0 and the associated auxiliary equation is r 2 + 4 r + 3 = 0 . Factorize r 2 + 4 r + 3 = 0 as, r 2 + 3 r + r + 3 = 0 r ( r + 3 ) + 1 ( r + 3 ) = 0 ( r + 1 ) ( r + 3 ) = 0 r = − 3 , − 1 That is, the roots are r 1 = − 3 and r 2 = − 1 . It is known that, if auxiliary equation has distinct real roots r 1 and r 2 , then y ( t ) = c 1 e r 1 t + c 2 e r 2 t is a general solution. Thus, the solution form y ( t ) = c 1 e r 1 t + c 2 e r 2 t becomes, y ( t ) = c 1 e − 3 t + c 2 e − t . That is, the general solution to the corresponding homogeneous equation of y ″ + 4 y ′ + 3 y = 9 is y h ( t ) = c 1 e − 3 t + c 2 e − t . Comparing y ″ + 4 y ′ + 3 y = 9 with a y ″ + b y ′ + c y = C t m e r t , it is observed that m = 0 and r = 0 . Since r = 0 is not a root of the associated auxiliary equation, s = 0 . The form y p ( t ) = t s ( A m t m + ⋯ + A 1 t + A 0 ) e r t with m = 0 , s = 0 and r = 0 will be, y p ( t ) = t 0 ( A 0 ) e 0 t = A 0 Differentiate y p ( t ) = A 0 with respect to t as, y ′ p ( t ) = 0 and y ″ p ( t ) = 0 . Thus, y ″ + 4 y ′ + 3 y for y p will be, y ″ p + 4 y ′ p + 3 y p = 0 + 4 ( 0 ) + 3 A 0 = 3 A 0 Setting 3 A 0 = 9 gives, A 0 = 3 . Thus, the particular solution y p ( t ) = A 0 becomes y p ( t ) = 3 . Combine the general solution y h ( t ) = c 1 e − 3 t + c 2 e − t and the particular solution y p ( t ) = 3 to obtain as, y ( t ) = c 1 e − 3 t + c 2 e − t + 3 . Given that, the system starts from rest. That is, y ′ ( 0 ) = y ( 0 ) = 0 . Apply the conditions y ( 0 ) = 0 and y ′ ( 0 ) = 0 to obtain the value of constants as follows: Substitute t = 0 and y ( 0 ) = 0 in y ( t ) = c 1 e − 3 t + c 2 e − t + 3 , y ( 0 ) = c 1 e − 3 × 0 + c 2 e 0 + 3 0 = c 1 ( 1 ) + c 2 ( 1 ) + 3 c 1 = − 3 − c 2 Differentiate y ( t ) = c 1 e − 3 t + c 2 e − t + 3 with respect to t as, y ′ ( t ) = − 3 c 1 e − 3 t − c 2 e − t . Substitute t = 0 and y ′ ( 0 ) = 0 in y ′ ( t ) = − 3 c 1 e − 3 t − c 2 e − t , y ′ ( 0 ) = − 3 c 1 e − 3 × 0 − c 2 e 0 0 = − 3 c 1 ( 1 ) − c 2 ( 1 ) c 2 = − 3 c 1 Use c 1 = − 3 − c 2 and simplify as, c 2 = − 3 ( − 3 − c 2 ) c 2 = 9 + 3 c 2 3 c 2 − c 2 = − 9 2 c 2 = − 9 c 2 = − 9 2 Thus, c 1 = − 3 − c 2 becomes, c 1 = − 3 − ( − 9 2 ) = 9 2 − 3 = 9 − 6 2 = 3 2 That is, c 1 = 3 2 and c 2 = − 9 2 . Thus, the solution y ( t ) = c 1 e − 3 t + c 2 e − t + 3 becomes y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 . Therefore, the displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 is y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 _ . Use online graphing calculator and sketch the graph of y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 as shown below in Figure 1. From Figure 1, it is observed that, the displacement y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 approaches 3 as t → ∞ . The limiting value of the displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 . The limiting value of the displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 is 3 _ . The displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 is y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 . Obatain the value of y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 as t → ∞ . lim t → ∞ y ( t ) = lim t → ∞ ( 3 2 e − 3 t − 9 2 e − t + 3 ) = 3 2 lim t → ∞ e − 3 t − 9 2 lim t → ∞ e − t + lim t → ∞ 3 = 3 2 lim b → ∞ e − b − 9 2 lim b → ∞ e − b + 3 = 3 2 ( 0 ) − 9 2 ( 0 ) + 3 ( lim b → ∞ e − b = 0 ) = 3 Therefore, the limiting value of the displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 is 3 _ . That is, the maximum displacement of the system is 3.

The displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 , where the system starts from rest and sketch the displacement y ( t ) . The displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 is y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 _ . Procedure used: Method of undetermined coefficients: A particular solution to the differential equation a y ″ + b y ′ + c y = C t m e r t , where m is a nonnegative integer, is of the form y p ( t ) = t s ( A m t m + ⋯ + A 1 t + A 0 ) e r t with: (i) s = 0 , if r is not a root of the associated auxiliary equation; (ii) s = 1 , if r is a simple root of the associated auxiliary equation; (iii) s = 2 , if r is a double root of the associated auxiliary equation. Calculation: Consider the system governed by the differential equation m d 2 y d t 2 + b d y d t + k y = F 0 cos γ t , where F 0 and γ are non-negative constants and b 2 > 4 m k . A solution to the above system has the form y = y h + y p , where y p is a particular solution and y h is a general solution to the corresponding homogeneous equation. Given that, m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 . Substitute m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 in m d 2 y d t 2 + b d y d t + k y = F 0 cos γ t , 2 d 2 y d t 2 + 8 d y d t + 6 y = 18 cos ( 0 × t ) 2 d 2 y d t 2 + 8 d y d t + 6 y = 18 cos ( 0 ) 2 d 2 y d t 2 + 8 d y d t + 6 y = 18 d 2 y d t 2 + 4 d y d t + 3 y = 9 ( divide by 2 on both sides ) That is, y ″ + 4 y ′ + 3 y = 9 . The corresponding homogenous equation is, y ″ + 4 y ′ + 3 y = 0 and the associated auxiliary equation is r 2 + 4 r + 3 = 0 . Factorize r 2 + 4 r + 3 = 0 as, r 2 + 3 r + r + 3 = 0 r ( r + 3 ) + 1 ( r + 3 ) = 0 ( r + 1 ) ( r + 3 ) = 0 r = − 3 , − 1 That is, the roots are r 1 = − 3 and r 2 = − 1 . It is known that, if auxiliary equation has distinct real roots r 1 and r 2 , then y ( t ) = c 1 e r 1 t + c 2 e r 2 t is a general solution. Thus, the solution form y ( t ) = c 1 e r 1 t + c 2 e r 2 t becomes, y ( t ) = c 1 e − 3 t + c 2 e − t . That is, the general solution to the corresponding homogeneous equation of y ″ + 4 y ′ + 3 y = 9 is y h ( t ) = c 1 e − 3 t + c 2 e − t . Comparing y ″ + 4 y ′ + 3 y = 9 with a y ″ + b y ′ + c y = C t m e r t , it is observed that m = 0 and r = 0 . Since r = 0 is not a root of the associated auxiliary equation, s = 0 . The form y p ( t ) = t s ( A m t m + ⋯ + A 1 t + A 0 ) e r t with m = 0 , s = 0 and r = 0 will be, y p ( t ) = t 0 ( A 0 ) e 0 t = A 0 Differentiate y p ( t ) = A 0 with respect to t as, y ′ p ( t ) = 0 and y ″ p ( t ) = 0 . Thus, y ″ + 4 y ′ + 3 y for y p will be, y ″ p + 4 y ′ p + 3 y p = 0 + 4 ( 0 ) + 3 A 0 = 3 A 0 Setting 3 A 0 = 9 gives, A 0 = 3 . Thus, the particular solution y p ( t ) = A 0 becomes y p ( t ) = 3 . Combine the general solution y h ( t ) = c 1 e − 3 t + c 2 e − t and the particular solution y p ( t ) = 3 to obtain as, y ( t ) = c 1 e − 3 t + c 2 e − t + 3 . Given that, the system starts from rest. That is, y ′ ( 0 ) = y ( 0 ) = 0 . Apply the conditions y ( 0 ) = 0 and y ′ ( 0 ) = 0 to obtain the value of constants as follows: Substitute t = 0 and y ( 0 ) = 0 in y ( t ) = c 1 e − 3 t + c 2 e − t + 3 , y ( 0 ) = c 1 e − 3 × 0 + c 2 e 0 + 3 0 = c 1 ( 1 ) + c 2 ( 1 ) + 3 c 1 = − 3 − c 2 Differentiate y ( t ) = c 1 e − 3 t + c 2 e − t + 3 with respect to t as, y ′ ( t ) = − 3 c 1 e − 3 t − c 2 e − t . Substitute t = 0 and y ′ ( 0 ) = 0 in y ′ ( t ) = − 3 c 1 e − 3 t − c 2 e − t , y ′ ( 0 ) = − 3 c 1 e − 3 × 0 − c 2 e 0 0 = − 3 c 1 ( 1 ) − c 2 ( 1 ) c 2 = − 3 c 1 Use c 1 = − 3 − c 2 and simplify as, c 2 = − 3 ( − 3 − c 2 ) c 2 = 9 + 3 c 2 3 c 2 − c 2 = − 9 2 c 2 = − 9 c 2 = − 9 2 Thus, c 1 = − 3 − c 2 becomes, c 1 = − 3 − ( − 9 2 ) = 9 2 − 3 = 9 − 6 2 = 3 2 That is, c 1 = 3 2 and c 2 = − 9 2 . Thus, the solution y ( t ) = c 1 e − 3 t + c 2 e − t + 3 becomes y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 . Therefore, the displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 is y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 _ . Use online graphing calculator and sketch the graph of y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 as shown below in Figure 1. From Figure 1, it is observed that, the displacement y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 approaches 3 as t → ∞ . The limiting value of the displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 . The limiting value of the displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 is 3 _ . The displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 is y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 . Obatain the value of y ( t ) = 3 2 e − 3 t − 9 2 e − t + 3 as t → ∞ . lim t → ∞ y ( t ) = lim t → ∞ ( 3 2 e − 3 t − 9 2 e − t + 3 ) = 3 2 lim t → ∞ e − 3 t − 9 2 lim t → ∞ e − t + lim t → ∞ 3 = 3 2 lim b → ∞ e − b − 9 2 lim b → ∞ e − b + 3 = 3 2 ( 0 ) − 9 2 ( 0 ) + 3 ( lim b → ∞ e − b = 0 ) = 3 Therefore, the limiting value of the displacement y ( t ) of the overdamped system with m = 2 , b = 8 , k = 6 , F 0 = 18 and γ = 0 is 3 _ . That is, the maximum displacement of the system is 3.

Solution Summary: The author calculates the displacement of the overdamped system with y(t).

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 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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