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21. Using Euler's formula, show that

21. Using Euler's formula, show that

Operations Research : Applications and Algorithms
Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter11: Nonlinear Programming
Section11.1: Review Of Differential Calculus
Problem 6P
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Therefore, e't = C₁ cost + C₂ sin t eir for some constants c₁ and c₂. Why is this so? c. Set t = 0 in equation (31) to show that C₁ = 1. d. Assuming that equation (15) is true, differentiate equation (31) and then set t = 0 to conclude that c₂ = i. Use the values of c₁ and c₂ in equation (31) to arrive at Euler's formula. 21. Using Euler's formula, show that eit + e-it 2 = cost, eit - e-it 2i = sin t. 22. If et is given by equation (14), show that e(+2) = e'¹¹ e 2¹ for any complex numbers r₁ and r2. 23. Consider the differential equation ay" + by' + cy = 0, where b² - 4ac < 0 and the characteristic equation has complex roots A ±iu. Substitute the functions u(t) = et cos(ut) and v(t) = et sin(ut) for y in the differential equation and thereby confirm that they are solutions. 24. If the functions y₁ and y2 are a fundamental set of solutions of y" + p(t) y'+q(t) y = 0, show that between consecutive zeros of y₁ there is one and only one zero of y2. Note that this result is illustrated by the solutions yi(t) = cost and y2(t) = sint of the equation y" + y = 0. 27. 12y 28. 12y 29. 12y 30. 1² 31. 12. Hint: Suppose that t₁ and t2 are two zeros of y₁ between which there are no zeros of y2. Apply Rolle's theorem to y₁/2 to reach a contradiction. Change of Variables. Sometimes a differential equation with variable coefficients, 32. In enable e coeffici y" + p(t)y' +q(t) y = 0, (32) can be put in a more suitable form for finding a solution by making a change of the independent variable. We explore these ideas in Problems 25 through 36. In particular, in Problem 25 we show that a class of equations known as Euler equations can be transformed independent variable. Problems 26 through 31 ara into equations with constant coefficients by a simple change of the the new specifie a. b. C I t
Transcribed Image Text:Therefore, e't = C₁ cost + C₂ sin t eir for some constants c₁ and c₂. Why is this so? c. Set t = 0 in equation (31) to show that C₁ = 1. d. Assuming that equation (15) is true, differentiate equation (31) and then set t = 0 to conclude that c₂ = i. Use the values of c₁ and c₂ in equation (31) to arrive at Euler's formula. 21. Using Euler's formula, show that eit + e-it 2 = cost, eit - e-it 2i = sin t. 22. If et is given by equation (14), show that e(+2) = e'¹¹ e 2¹ for any complex numbers r₁ and r2. 23. Consider the differential equation ay" + by' + cy = 0, where b² - 4ac < 0 and the characteristic equation has complex roots A ±iu. Substitute the functions u(t) = et cos(ut) and v(t) = et sin(ut) for y in the differential equation and thereby confirm that they are solutions. 24. If the functions y₁ and y2 are a fundamental set of solutions of y" + p(t) y'+q(t) y = 0, show that between consecutive zeros of y₁ there is one and only one zero of y2. Note that this result is illustrated by the solutions yi(t) = cost and y2(t) = sint of the equation y" + y = 0. 27. 12y 28. 12y 29. 12y 30. 1² 31. 12. Hint: Suppose that t₁ and t2 are two zeros of y₁ between which there are no zeros of y2. Apply Rolle's theorem to y₁/2 to reach a contradiction. Change of Variables. Sometimes a differential equation with variable coefficients, 32. In enable e coeffici y" + p(t)y' +q(t) y = 0, (32) can be put in a more suitable form for finding a solution by making a change of the independent variable. We explore these ideas in Problems 25 through 36. In particular, in Problem 25 we show that a class of equations known as Euler equations can be transformed independent variable. Problems 26 through 31 ara into equations with constant coefficients by a simple change of the the new specifie a. b. C I t
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