a. Draw a direction field for the differential equation (or reexamine the one from Problem 7). Observe that there is a critical value of a in the interval 2 ≤ a ≤ 3 that separates converging solutions from diverging ones. Call this critical value ao. N b. Use Euler's method with h = 0.01 to estimate ao. Do this by restricting a to an interval [a, b], where b - a = 0.01.
a. Draw a direction field for the differential equation (or reexamine the one from Problem 7). Observe that there is a critical value of a in the interval 2 ≤ a ≤ 3 that separates converging solutions from diverging ones. Call this critical value ao. N b. Use Euler's method with h = 0.01 to estimate ao. Do this by restricting a to an interval [a, b], where b - a = 0.01.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
13
![1.
2.
3.
c. Repeat part a with h = 0.025. Compare the results with
those found in a and b.
(Estoq
Nd. Find the solution y = (1) of the given problem and
evaluate (1) at t = 0.1, 0.2, 0.3, and 0.4. Compare these values
with the results of a, b, and c.
y'=3+t-y, y(0) = 1
y' = 2y = 1, y(0) = 1
y'=0.5-1+2y,
y(0) = 1 or
co
4. y'= 3 cost-2y, y(0) = 0
-
G
In each of Problems 5 through 8, draw a direction field for the given
differential equation and state whether you think that the solutions are
converging or diverging.
st
Uns
vad
-
5. y' = 5-3√//y
G
6.
y' = y(3-ty)
G 7. y'= -ty+0.1y³
G 8. y'= 1² + y² ni maldong
y':
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
31²
=
aldong oft sh
w stanicere 3y² - 4' y(1) = 0.
where a is a given number.
Heuni
Ona
2.8 The Existence and Uniqueness Theorem
a. Draw a direction field for the differential equation (or
reexamine the one from Problem 7). Observe that there is a
critical value of a in the interval 2 ≤ a ≤ 3 that separates
tom on converging solutions from diverging ones. Call this critical
par
pindlagt no fing abr
euluolto besanvin no setoo si ni oni r
1
that separates converging solutions from diverging ones. Call this
critical value ao.
sisolbai svawor,ob sw road toong b. Use Euler's method with h = 0.01 to estimate ao. Do this
eich.
smobravo od 12mm sari
by restricting ao to an interval [a, b], where b-a = 0.01.
15. Convergence of Euler's Method. It can be shown that
under suitable conditions on f, the numerical approximation
generated by the Euler method for the initial value problem
y' = f(t, y), y(to) = yo converges to the exact solution as the step
size h decreases. This is illustrated by the following example. Consider
the initial value problem
y = 1-t+y, y(to) = yo.
14.
Jon ob
(1) 0
value ao.
N b. Use Euler's method with h = 0.01 to estimate ao. Do this
by restricting ao to an interval [a, b], where b - a = 0.01.
Consider the initial value problem
y = y² - t², y(0) = a,
sonsunez ar to prodmam
where a is a given number.
WE
Ga. Draw a direction field for the differential equation. Note
that there is a critical value of a in the interval 0 ≤ a ≤ 1
the process produ
Na. Use Euler's method with h = 0.1 to obtain approximate oulunos 10
be
values 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 are
are ib toord to bodism
to bodism orli Yn = (1 + h)" (yo-to) + tn
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
d. Consider a fixed point t > to and for a given n choose
h = (t-to)/n. Then tn = t for every n. Note also that h→0
gainis as n → ∞o. By substituting for h in equation (19) and letting
n→ ∞, show that yn → (t) as n → ∞.
83
a. Show that the exact solution is y = o(t) = (yo-to) e ¹0+t.
N b. Using the Euler formula, show that
yk = (1 + h) yk-1+h-htk-1, k = 1, 2, ....
6116, boe
c. Noting that y₁ = (1 + h) (yoto) + t₁, show by induction
that
1-
• noitibaoo labsinton In each of Problems 16 and 17, use the technique discussed in Problem
15 to show that the approximation obtained by the Euler method
converges to the exact solution at any fixed point as h→0.
Hint: lim (1+a/n)" = eª.
818
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 oil 16. y' = y, y(0) = 1
consistent with the direction field in Problem 8?
13. Consider the initial value problem
y' = -ty+0.1y³,
17. y' = 2y1, y(0) = 1
LHIDO
y(0) = a,
laikiodi asitaire qals noltanut aids nodi
100 ishini sdi jadi awode daidw (E) qoilaups ni 1 zot orss
1241 most rellot 11 avoit (E) noileups ni bargaini
aft (0)6.1)1
1- (1) tadi bas oldalidsnettib zijn:
(19)
Hint: y₁ = (1+2h)/2+1/2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe58a88cd-59d9-4958-9d94-9dac5f80a3f7%2Fe3277639-0e28-4c9c-8bd2-9f3522eeb2f2%2Fgmi7pu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.
2.
3.
c. Repeat part a with h = 0.025. Compare the results with
those found in a and b.
(Estoq
Nd. Find the solution y = (1) of the given problem and
evaluate (1) at t = 0.1, 0.2, 0.3, and 0.4. Compare these values
with the results of a, b, and c.
y'=3+t-y, y(0) = 1
y' = 2y = 1, y(0) = 1
y'=0.5-1+2y,
y(0) = 1 or
co
4. y'= 3 cost-2y, y(0) = 0
-
G
In each of Problems 5 through 8, draw a direction field for the given
differential equation and state whether you think that the solutions are
converging or diverging.
st
Uns
vad
-
5. y' = 5-3√//y
G
6.
y' = y(3-ty)
G 7. y'= -ty+0.1y³
G 8. y'= 1² + y² ni maldong
y':
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
31²
=
aldong oft sh
w stanicere 3y² - 4' y(1) = 0.
where a is a given number.
Heuni
Ona
2.8 The Existence and Uniqueness Theorem
a. Draw a direction field for the differential equation (or
reexamine the one from Problem 7). Observe that there is a
critical value of a in the interval 2 ≤ a ≤ 3 that separates
tom on converging solutions from diverging ones. Call this critical
par
pindlagt no fing abr
euluolto besanvin no setoo si ni oni r
1
that separates converging solutions from diverging ones. Call this
critical value ao.
sisolbai svawor,ob sw road toong b. Use Euler's method with h = 0.01 to estimate ao. Do this
eich.
smobravo od 12mm sari
by restricting ao to an interval [a, b], where b-a = 0.01.
15. Convergence of Euler's Method. It can be shown that
under suitable conditions on f, the numerical approximation
generated by the Euler method for the initial value problem
y' = f(t, y), y(to) = yo converges to the exact solution as the step
size h decreases. This is illustrated by the following example. Consider
the initial value problem
y = 1-t+y, y(to) = yo.
14.
Jon ob
(1) 0
value ao.
N b. Use Euler's method with h = 0.01 to estimate ao. Do this
by restricting ao to an interval [a, b], where b - a = 0.01.
Consider the initial value problem
y = y² - t², y(0) = a,
sonsunez ar to prodmam
where a is a given number.
WE
Ga. Draw a direction field for the differential equation. Note
that there is a critical value of a in the interval 0 ≤ a ≤ 1
the process produ
Na. Use Euler's method with h = 0.1 to obtain approximate oulunos 10
be
values 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 are
are ib toord to bodism
to bodism orli Yn = (1 + h)" (yo-to) + tn
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
d. Consider a fixed point t > to and for a given n choose
h = (t-to)/n. Then tn = t for every n. Note also that h→0
gainis as n → ∞o. By substituting for h in equation (19) and letting
n→ ∞, show that yn → (t) as n → ∞.
83
a. Show that the exact solution is y = o(t) = (yo-to) e ¹0+t.
N b. Using the Euler formula, show that
yk = (1 + h) yk-1+h-htk-1, k = 1, 2, ....
6116, boe
c. Noting that y₁ = (1 + h) (yoto) + t₁, show by induction
that
1-
• noitibaoo labsinton In each of Problems 16 and 17, use the technique discussed in Problem
15 to show that the approximation obtained by the Euler method
converges to the exact solution at any fixed point as h→0.
Hint: lim (1+a/n)" = eª.
818
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 oil 16. y' = y, y(0) = 1
consistent with the direction field in Problem 8?
13. Consider the initial value problem
y' = -ty+0.1y³,
17. y' = 2y1, y(0) = 1
LHIDO
y(0) = a,
laikiodi asitaire qals noltanut aids nodi
100 ishini sdi jadi awode daidw (E) qoilaups ni 1 zot orss
1241 most rellot 11 avoit (E) noileups ni bargaini
aft (0)6.1)1
1- (1) tadi bas oldalidsnettib zijn:
(19)
Hint: y₁ = (1+2h)/2+1/2
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