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CalculusCalculus: Early TranscendentalsLet L be a nonzero real number. (a) Show that the boundary-value problem y" + λ y = 0, y (0) = 0, y ( L ) = 0 has only the trivial solution y = 0 for the cases λ = C and λ < 0. (b) For the case λ > 0, find the values of λ for which this problem has a nontrivial solution and give the corresponding solution.BuyFind*launch*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285741550

Chapter 17.1, Problem 33E

Textbook Problem

Let *L* be a nonzero real number.

(a) Show that the boundary-value problem *y"* + λ*y* = 0, *y*(0) = 0, *y*(*L*) = 0 has only the trivial solution *y* = 0 for the cases λ = C and λ < 0.

(b) For the case λ > 0, find the values of λ for which this problem has a nontrivial solution and give the corresponding solution.

Calculus: Early Transcendentals

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Ch. 17.1 - Solve the differential equation. 1. y" y' 6y = 0Ch. 17.1 - Solve the differential equation. 2. y" 6y' + 9y =...Ch. 17.1 - Solve the differential equation. 3. y" + 2y = 0Ch. 17.1 - Solve the differential equation. 4. y" + y' 12y =...Ch. 17.1 - Solve the differential equation. 5. 4y" + 4y' + y...Ch. 17.1 - Solve the differential equation. 6. 9y" + 4y = 0Ch. 17.1 - Solve the differential equation. 7. 3y" = 4y'Ch. 17.1 - Solve the differential equation. 8. y = y"Ch. 17.1 - Solve the differential equation. 9. y" 4y' + 13y...Ch. 17.1 - Solve the differential equation. 10. 3y" + 4y' 3y...

Ch. 17.1 - Solve the differential equation. 11....Ch. 17.1 - Solve the differential equation. 12....Ch. 17.1 - Solve the differential equation. 13....Ch. 17.1 - Graph the two basic solutions along with several...Ch. 17.1 - Graph the two basic solutions along with several...Ch. 17.1 - Graph the two basic solutions along with several...Ch. 17.1 - Solve the initial-value problem. 17. y" + 3y = 0,...Ch. 17.1 - Solve the initial-value problem. 18. y" - 2y' 3y...Ch. 17.1 - Solve the initial-value problem. 19. 9y" + 12y' +...Ch. 17.1 - Solve the initial-value problem. 20. 3y" 2y' y =...Ch. 17.1 - Solve the initial-value problem. 21. y" 6y' + 10y...Ch. 17.1 - Solve the initial-value problem. 22. 4y" 20y' +...Ch. 17.1 - Solve the initial-value problem. 23. y" y' 12y =...Ch. 17.1 - Solve the initial-value problem. 24. 4y" + 4y' +...Ch. 17.1 - Solve the boundary-value problem, if possible. 25....Ch. 17.1 - Solve the boundary-value problem, if possible. 26....Ch. 17.1 - Solve the boundary-value problem, if possible. 27....Ch. 17.1 - Solve the boundary-value problem, if possible. 28....Ch. 17.1 - Solve the boundary-value problem, if possible. 29....Ch. 17.1 - Solve the boundary-value problem, if possible. 30....Ch. 17.1 - Solve the boundary-value problem, if possible. 31....Ch. 17.1 - Solve the boundary-value problem, if possible. 32....Ch. 17.1 - Let L be a nonzero real number. (a) Show that the...Ch. 17.1 - If a, b, and c are all positive constants and y(x)...Ch. 17.2 - Solve the differential equation or initial-value...Ch. 17.2 - Solve the differential equation or initial-value...Ch. 17.2 - Solve the differential equation or initial-value...Ch. 17.2 - Solve the differential equation or initial-value...Ch. 17.2 - Solve the differential equation or initial-value...Ch. 17.2 - Solve the differential equation or initial-value...Ch. 17.2 - Solve the differential equation or initial-value...Ch. 17.2 - Solve the differential equation or initial-value...Ch. 17.2 - Solve the differential equation or initial-value...Ch. 17.2 - Solve the differential equation or initial-value...Ch. 17.2 - Graph the particular solution and several other...Ch. 17.2 - Graph the particular solution and several other...Ch. 17.2 - Write a trial solution for the method of...Ch. 17.2 - Write a trial solution for the method of...Ch. 17.2 - Write a trial solution for the method of...Ch. 17.2 - Write a trial solution for the method of...Ch. 17.2 - Write a trial solution for the method of...Ch. 17.2 - Write a trial solution for the method of...Ch. 17.2 - Solve the differential equation using (a)...Ch. 17.2 - Solve the differential equation using (a)...Ch. 17.2 - Solve the differential equation using (a)...Ch. 17.2 - Solve the differential equation using (a)...Ch. 17.2 - Solve the differential equation using the method...Ch. 17.2 - Solve the differential equation using the method...Ch. 17.2 - Solve the differential equation using the method...Ch. 17.2 - Solve the differential equation using the method...Ch. 17.2 - Solve the differential equation using the method...Ch. 17.2 - Solve the differential equation using the method...Ch. 17.3 - A spring has natural length 0.75 m and a 5-kg...Ch. 17.3 - A spring with an 8-kg mass is kept stretched 0.4 m...Ch. 17.3 - A spring with a mass of 2 kg has damping constant...Ch. 17.3 - A force of 13 N is needed to keep a spring with a...Ch. 17.3 - For the spring in Exercise 3, find the mass that...Ch. 17.3 - For the spring in Exercise 4, find the damping...Ch. 17.3 - A spring has a mass of 1 kg and its spring...Ch. 17.3 - A spring has a mass of 1 kg and its damping...Ch. 17.3 - Suppose a spring has mass m and spring constant k...Ch. 17.3 - As in Exercise 9, consider a spring with mass m,...Ch. 17.3 - Show that if 0 , but /0 is a rational number,...Ch. 17.3 - Consider a spring subject to a frictional or...Ch. 17.3 - A series circuit consists of a resistor with R =...Ch. 17.3 - A series circuit contains a resistor with R = 24 ,...Ch. 17.3 - The battery in Exercise 13 is replaced by a...Ch. 17.3 - The battery in Exercise 14 is replaced by a...Ch. 17.3 - Verify that the solution to Equation I can be...Ch. 17.3 - The figure shows a pendulum with length I, and the...Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - 1-11 Use power series to solve the differential...Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - The solution of the initial-value problem x2y" +...Ch. 17 - (a) Write the general form of a second-order...Ch. 17 - (a) What is an initial-value problem for a...Ch. 17 - (a) Write the general form of a second-order...Ch. 17 - Discuss two applications of second-order linear...Ch. 17 - How do you use power series to solve a...Ch. 17 - Determine whether the statement is true or false....Ch. 17 - Determine whether the statement is true or false....Ch. 17 - Determine whether the statement is true or false....Ch. 17 - Determine whether the statement is true or false....Ch. 17 - Solve the differential equation. 1. 4y" y =0Ch. 17 - Solve the differential equation. 2. 2y" 2y' + 10y...Ch. 17 - Solve the differential equation. 3. y" + 3y =0Ch. 17 - Solve the differential equation. 4. y" + 8y' + 16y...Ch. 17 - Solve the differential equation. 5....Ch. 17 - Solve the differential equation. 6....Ch. 17 - Solve the differential equation. 7....Ch. 17 - Solve the differential equation. 8....Ch. 17 - Solve the differential equation. 9....Ch. 17 - Solve the differential equation. 10....Ch. 17 - Solve the initial-value problem. 11. y" + 6y' = 0,...Ch. 17 - Solve the initial-value problem. 12. y" 6y' + 25y...Ch. 17 - Solve the initial-value problem. 13. y" 5y' + 4y...Ch. 17 - Solve the initial-value problem. 14. 9y" + y =3x +...Ch. 17 - Solve the boundary-value problem, if possible. 15....Ch. 17 - Solve the boundary-value problem, if possible. 16....Ch. 17 - Use power series to solve the initial-value...Ch. 17 - Use power series to solve differential equation y"...Ch. 17 - A series circuit contains a resistor with R = 40 ,...Ch. 17 - A spring with a mass of 2 kg has damping constant...Ch. 17 - Assume that the earth is a solid sphere of uniform...

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