Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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9a
converging or diverging. 5. y' = 5-3√y 6. y' = y(3-ty) 7. y'= -ty+0.1y³ 8. G G G In each of Problems 9 and 10, use Euler's method to find approximate values of the solution of the given initial value problem at t = 0.5, 1, 1.5, 2, 2.5, and 3: (a) With h = 0.1, (b) With h = 0.05, (c) With h = 0.025, (d) With h = 0.01. N 9. y'=5-3√y, y(0) = 2 N 10. y'= y(3-ty), y(0) = 0.5 11. Consider the initial value problem y' = Lins zsupinted o Rea yyy y'= 1² + y² ni maldong 31² 3y24 Na. Use Euler's method with h = 0.1 to obtain approximate values of the solution at t = 1.2, 1.4, 1.6, and 1.8. N b. Repeat part a with h = 0.05. c. Compare the results of parts a and b. Note that they are reasonably close for t = 1.2, 1.4, and 1.6 but are quite different for t= 1.8. Also note (from the differential equation) that the line tangent to the solution is parallel to the y-axis when y = ±2/√√3 ±1.155. Explain how this might cause such a difference in the calculated values. N 12. Consider the initial value problem y' = 1² + y², y(0) = 1. Use Euler's method with h = 0.1, 0.05, 0.025, and 0.01 to explore the solution of this problem for 0 < t < 1. What is your best estimate of the value of the solution at t = 0.8? At t = 1? Are your results consistent with the direction field in Problem 8? modsups y(1) = 0. 13. Consider the initial value problem fatui 2.8 where a is a given number. y = -ty+0.1y³, y(0) = a, that there is a critical that separates convergir critical value ao. N b. Use Euler's meth by restricting ao to an 15. Convergence of EL under suitable conditions generated by the Euler y' = f(t, y), y(to) = yo size h decreases. This is illu the initial value problem y' = 1 a. Show that the exa N b. Using the Eul. Yk= (1 + h) c. Noting that y₁ = that on si Ya for each positive in d. Consider a fixe h=(t-to)/n. T as n →∞o. By su n→ ∞, show tha Hint: lim (1+a/ 84x In each of Problems 16 15 to show that the a converges to the exact 16. y'= y, y(0) = 17. y' = 2y - 1, anini The Existence and Uniqueness Theor idh aoran al hor In this section we discuss the proof of Theorem 2.4.2, the fundamental exister uniqueness theorem for first-order initial value problems. Recall that this theorem sta conditions on f(t. y), the initial value problem under contain pomp
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Transcribed Image Text:converging or diverging. 5. y' = 5-3√y 6. y' = y(3-ty) 7. y'= -ty+0.1y³ 8. G G G In each of Problems 9 and 10, use Euler's method to find approximate values of the solution of the given initial value problem at t = 0.5, 1, 1.5, 2, 2.5, and 3: (a) With h = 0.1, (b) With h = 0.05, (c) With h = 0.025, (d) With h = 0.01. N 9. y'=5-3√y, y(0) = 2 N 10. y'= y(3-ty), y(0) = 0.5 11. Consider the initial value problem y' = Lins zsupinted o Rea yyy y'= 1² + y² ni maldong 31² 3y24 Na. Use Euler's method with h = 0.1 to obtain approximate values of the solution at t = 1.2, 1.4, 1.6, and 1.8. N b. Repeat part a with h = 0.05. c. Compare the results of parts a and b. Note that they are reasonably close for t = 1.2, 1.4, and 1.6 but are quite different for t= 1.8. Also note (from the differential equation) that the line tangent to the solution is parallel to the y-axis when y = ±2/√√3 ±1.155. Explain how this might cause such a difference in the calculated values. N 12. Consider the initial value problem y' = 1² + y², y(0) = 1. Use Euler's method with h = 0.1, 0.05, 0.025, and 0.01 to explore the solution of this problem for 0 < t < 1. What is your best estimate of the value of the solution at t = 0.8? At t = 1? Are your results consistent with the direction field in Problem 8? modsups y(1) = 0. 13. Consider the initial value problem fatui 2.8 where a is a given number. y = -ty+0.1y³, y(0) = a, that there is a critical that separates convergir critical value ao. N b. Use Euler's meth by restricting ao to an 15. Convergence of EL under suitable conditions generated by the Euler y' = f(t, y), y(to) = yo size h decreases. This is illu the initial value problem y' = 1 a. Show that the exa N b. Using the Eul. Yk= (1 + h) c. Noting that y₁ = that on si Ya for each positive in d. Consider a fixe h=(t-to)/n. T as n →∞o. By su n→ ∞, show tha Hint: lim (1+a/ 84x In each of Problems 16 15 to show that the a converges to the exact 16. y'= y, y(0) = 17. y' = 2y - 1, anini The Existence and Uniqueness Theor idh aoran al hor In this section we discuss the proof of Theorem 2.4.2, the fundamental exister uniqueness theorem for first-order initial value problems. Recall that this theorem sta conditions on f(t. y), the initial value problem under contain pomp
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