The approximate values of the solution of initial value problem y ′ = 2 y − 1 , y ( 0 ) = 1 using Euler method with h = 0.1 at t = 0.1 , 0 .2, 0 .3, and 0 .4 .

Question

6. Show that 1{AU B} = max{1{A}, I{B}} = I{A} + I{B} - I{A} I{B}; I{AB} = min{I{A}, I{B}} = I{A} I{B}; I{A A B} = I{A} + I{B}-21{A} I {B} = (I{A} - I{B})².

Answer 1

Question

Chapter 2.7, Problem 2P

(a)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.1 at t=0.1, 0.2, 0.3, and 0.4.

(b)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.05 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a).

(c)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.025 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a) and part (b).

(d)

To determine

The solution y=ϕ(t) and values of ϕ(t) at t=0.1, 0.2, 0.3, and 0.4. Also compare the results with those obtained in part (a), (b), and (c).

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6. Show that 1{AU B} = max{1{A}, I{B}} = I{A} + I{B} - I{A} I{B}; I{AB} = min{I{A}, I{B}} = I{A} I{B}; I{A A B} = I{A} + I{B}-21{A} I {B} = (I{A} - I{B})².

Theorem 3.5 Suppose that P and Q are probability measures defined on the same probability space (2, F), and that F is generated by a л-system A. If P(A) = Q(A) for all A = A, then P = Q, i.e., P(A) = Q(A) for all A = F.

6. Show that, for any random variable, X, and a > 0, Lo P(x -00 P(x < x

Answer 2

Question

Chapter 2.7, Problem 2P

(a)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.1 at t=0.1, 0.2, 0.3, and 0.4.

(b)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.05 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a).

(c)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.025 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a) and part (b).

(d)

To determine

The solution y=ϕ(t) and values of ϕ(t) at t=0.1, 0.2, 0.3, and 0.4. Also compare the results with those obtained in part (a), (b), and (c).

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Answer 3

Question

Chapter 2.7, Problem 2P

(a)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.1 at t=0.1, 0.2, 0.3, and 0.4.

(b)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.05 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a).

(c)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.025 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a) and part (b).

(d)

To determine

The solution y=ϕ(t) and values of ϕ(t) at t=0.1, 0.2, 0.3, and 0.4. Also compare the results with those obtained in part (a), (b), and (c).

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Answer 4

Question

Chapter 2.7, Problem 2P

(a)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.1 at t=0.1, 0.2, 0.3, and 0.4.

(b)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.05 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a).

(c)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.025 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a) and part (b).

(d)

To determine

The solution y=ϕ(t) and values of ϕ(t) at t=0.1, 0.2, 0.3, and 0.4. Also compare the results with those obtained in part (a), (b), and (c).

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Answer 5

Chapter 2.7, Problem 2P

(a)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.1 at t=0.1, 0.2, 0.3, and 0.4.

(b)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.05 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a).

(c)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.025 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a) and part (b).

(d)

To determine

The solution y=ϕ(t) and values of ϕ(t) at t=0.1, 0.2, 0.3, and 0.4. Also compare the results with those obtained in part (a), (b), and (c).

Answer 6

Chapter 2.7, Problem 2P

(a)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.1 at t=0.1, 0.2, 0.3, and 0.4.

Answer 7

Chapter 2.7, Problem 2P

(a)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.1 at t=0.1, 0.2, 0.3, and 0.4.

Answer 8

(b)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.05 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a).

Answer 9

(b)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.05 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a).

Answer 10

(c)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.025 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a) and part (b).

Answer 11

(c)

To determine

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.025 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a) and part (b).

Answer 12

(d)

To determine

The solution y=ϕ(t) and values of ϕ(t) at t=0.1, 0.2, 0.3, and 0.4. Also compare the results with those obtained in part (a), (b), and (c).

Answer 13

(d)

To determine

The solution y=ϕ(t) and values of ϕ(t) at t=0.1, 0.2, 0.3, and 0.4. Also compare the results with those obtained in part (a), (b), and (c).

Answer 14

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Ch. 2.1 - Prob. 1P Ch. 2.1 - Prob. 2P Ch. 2.1 - Prob. 3P Ch. 2.1 - Prob. 4P Ch. 2.1 - Prob. 5P Ch. 2.1 - Prob. 6P Ch. 2.1 - Prob. 7P Ch. 2.1 - Prob. 8P Ch. 2.1 - Prob. 9P Ch. 2.1 - Prob. 10P

The approximate values of the solution of initial value problem y ′ = 2 y − 1 , y ( 0 ) = 1 using Euler method with h = 0.1 at t = 0.1 , 0 .2, 0 .3, and 0 .4 .

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.1 at t=0.1, 0.2, 0.3, and 0.4.

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.05 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a).

The approximate values of the solution of initial value problem y′=2y−1, y(0)=1 using Euler method with h=0.025 at t=0.1, 0.2, 0.3, and 0.4 and compare the results with those obtained in part (a) and part (b).

The solution y=ϕ(t) and values of ϕ(t) at t=0.1, 0.2, 0.3, and 0.4. Also compare the results with those obtained in part (a), (b), and (c).

Chapter 2 Solutions