13 1.2.3.c. Repeat part a with h = 0.025. Compare the results withthose found in a and b.(EstoqNd. Find the solution y = (1) of the given problem andevaluate (1) at t = 0.1, 0.2, 0.3, and 0.4. Compare these valueswith the results of a, b, and c.y'=3+t-y, y(0) = 1y' = 2y = 1, y(0) = 1y'=0.5-1+2y,y(0) = 1 orco4. y'= 3 cost-2y, y(0) = 0-GIn each of Problems 5 through 8, draw a direction field for the givendifferential equation and state whether you think that the solutions areconverging or diverging.stUnsvad-5. y' = 5-3√//yG6.y' = y(3-ty)G 7. y'= -ty+0.1y³G 8. y'= 1² + y² ni maldongy':In each of Problems 9 and 10, use Euler's method to find approximatevalues 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) Withh = 0.025, (d) With h = 0.01.N 9. y'=5-3√y,y(0) = 2N 10. y'= y(3-ty), y(0) = 0.511. Consider the initial value problem31²=aldong oft shw stanicere 3y² - 4' y(1) = 0.where a is a given number.HeuniOna2.8 The Existence and Uniqueness Theorema. Draw a direction field for the differential equation (orreexamine the one from Problem 7). Observe that there is acritical value of a in the interval 2 ≤ a ≤ 3 that separatestom on converging solutions from diverging ones. Call this criticalparpindlagt no fing abreuluolto besanvin no setoo si ni oni r1that separates converging solutions from diverging ones. Call thiscritical value ao.sisolbai svawor,ob sw road toong b. Use Euler's method with h = 0.01 to estimate ao. Do thiseich.smobravo od 12mm sariby restricting ao to an interval [a, b], where b-a = 0.01.15. Convergence of Euler's Method. It can be shown thatunder suitable conditions on f, the numerical approximationgenerated by the Euler method for the initial value problemy' = f(t, y), y(to) = yo converges to the exact solution as the stepsize h decreases. This is illustrated by the following example. Considerthe initial value problemy = 1-t+y, y(to) = yo.14.Jon ob(1) 0value ao.N b. Use Euler's method with h = 0.01 to estimate ao. Do thisby restricting ao to an interval [a, b], where b - a = 0.01.Consider the initial value problemy = y² - t², y(0) = a,sonsunez ar to prodmamwhere a is a given number.WEGa. Draw a direction field for the differential equation. Notethat there is a critical value of a in the interval 0 ≤ a ≤ 1the process produNa. Use Euler's method with h = 0.1 to obtain approximate oulunos 10bevalues of the solution at t = 1.2, 1.4, 1.6, and 1.8.Nb. Repeat part a with h = 0.05.Col moin for each positive integer n.c. Compare the results of parts a and b. Note that they areare ib toord to bodismto bodism orli Yn = (1 + h)" (yo-to) + tnreasonably close for t = 1.2, 1.4, and 1.6 but are quite differentfor t= 1.8. Also note (from the differential equation) thatthe line tangent to the solution is parallel to the y-axis wheny = +2/√3 ±1.155. Explain how this might cause sucha difference in the calculated values.N 12. Consider the initial value problemd. Consider a fixed point t to and for a given n chooseh = (t-to)/n. Then tn = t for every n. Note also that h→0gainis as n → ∞o. By substituting for h in equation (19) and lettingn→ ∞, show that yn → (t) as n → ∞.83a. Show that the exact solution is y = o(t) = (yo-to) e ¹0+t.N b. Using the Euler formula, show thatyk = (1 + h) yk-1+h-htk-1, k = 1, 2, ....6116, boec. Noting that y₁ = (1 + h) (yoto) + t₁, show by inductionthat1-• noitibaoo labsinton In each of Problems 16 and 17, use the technique discussed in Problem15 to show that the approximation obtained by the Euler methodconverges to the exact solution at any fixed point as h→0.Hint: lim (1+a/n)" = eª.818y' = 1² + y², y(0) = 1.Use Euler's method with h = 0.1, 0.05, 0.025, and 0.01 to explore thesolution of this problem for 0

Question

13 1.2.3.c. Repeat part a with h = 0.025. Compare the results withthose found in a and b.(EstoqNd. Find the solution y = (1) of the given problem andevaluate (1) at t = 0.1, 0.2, 0.3, and 0.4. Compare these valueswith the results of a, b, and c.y'=3+t-y, y(0) = 1y' = 2y = 1, y(0) = 1y'=0.5-1+2y,y(0) = 1 orco4. y'= 3 cost-2y, y(0) = 0-GIn each of Problems 5 through 8, draw a direction field for the givendifferential equation and state whether you think that the solutions areconverging or diverging.stUnsvad-5. y' = 5-3√//yG6.y' = y(3-ty)G 7. y'= -ty+0.1y³G 8. y'= 1² + y² ni maldongy':In each of Problems 9 and 10, use Euler's method to find approximatevalues 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) Withh = 0.025, (d) With h = 0.01.N 9. y'=5-3√y,y(0) = 2N 10. y'= y(3-ty), y(0) = 0.511. Consider the initial value problem31²=aldong oft shw stanicere 3y² - 4' y(1) = 0.where a is a given number.HeuniOna2.8 The Existence and Uniqueness Theorema. Draw a direction field for the differential equation (orreexamine the one from Problem 7). Observe that there is acritical value of a in the interval 2 ≤ a ≤ 3 that separatestom on converging solutions from diverging ones. Call this criticalparpindlagt no fing abreuluolto besanvin no setoo si ni oni r1that separates converging solutions from diverging ones. Call thiscritical value ao.sisolbai svawor,ob sw road toong b. Use Euler's method with h = 0.01 to estimate ao. Do thiseich.smobravo od 12mm sariby restricting ao to an interval [a, b], where b-a = 0.01.15. Convergence of Euler's Method. It can be shown thatunder suitable conditions on f, the numerical approximationgenerated by the Euler method for the initial value problemy' = f(t, y), y(to) = yo converges to the exact solution as the stepsize h decreases. This is illustrated by the following example. Considerthe initial value problemy = 1-t+y, y(to) = yo.14.Jon ob(1) 0value ao.N b. Use Euler's method with h = 0.01 to estimate ao. Do thisby restricting ao to an interval [a, b], where b - a = 0.01.Consider the initial value problemy = y² - t², y(0) = a,sonsunez ar to prodmamwhere a is a given number.WEGa. Draw a direction field for the differential equation. Notethat there is a critical value of a in the interval 0 ≤ a ≤ 1the process produNa. Use Euler's method with h = 0.1 to obtain approximate oulunos 10bevalues of the solution at t = 1.2, 1.4, 1.6, and 1.8.Nb. Repeat part a with h = 0.05.Col moin for each positive integer n.c. Compare the results of parts a and b. Note that they areare ib toord to bodismto bodism orli Yn = (1 + h)&#34; (yo-to) + tnreasonably close for t = 1.2, 1.4, and 1.6 but are quite differentfor t= 1.8. Also note (from the differential equation) thatthe line tangent to the solution is parallel to the y-axis wheny = +2/√3 ±1.155. Explain how this might cause sucha difference in the calculated values.N 12. Consider the initial value problemd. Consider a fixed point t > to and for a given n chooseh = (t-to)/n. Then tn = t for every n. Note also that h→0gainis as n → ∞o. By substituting for h in equation (19) and lettingn→ ∞, show that yn → (t) as n → ∞.83a. Show that the exact solution is y = o(t) = (yo-to) e ¹0+t.N b. Using the Euler formula, show thatyk = (1 + h) yk-1+h-htk-1, k = 1, 2, ....6116, boec. Noting that y₁ = (1 + h) (yoto) + t₁, show by inductionthat1-• noitibaoo labsinton In each of Problems 16 and 17, use the technique discussed in Problem15 to show that the approximation obtained by the Euler methodconverges to the exact solution at any fixed point as h→0.Hint: lim (1+a/n)&#34; = eª.818y' = 1² + y², y(0) = 1.Use Euler's method with h = 0.1, 0.05, 0.025, and 0.01 to explore thesolution of this problem for 0 < t < 1. What is your best estimateof the value of the solution at t = 0.8? At t = 1? Are your results oil 16. y' = y, y(0) = 1consistent with the direction field in Problem 8?13. Consider the initial value problemy' = -ty+0.1y³,17. y' = 2y1, y(0) = 1LHIDOy(0) = a,laikiodi asitaire qals noltanut aids nodi100 ishini sdi jadi awode daidw (E) qoilaups ni 1 zot orss1241 most rellot 11 avoit (E) noileups ni bargainiaft (0)6.1)11- (1) tadi bas oldalidsnettib zijn:(19)Hint: y₁ = (1+2h)/2+1/2

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