Explanation Given: Generalized Blasius Equation H. Blasius, in his study of laminar flow of a fluid, encountered an equation of the form y ′′ ′ + y y ′′ = ( y ′ ) 2 − 1 with initial conditions y ( 0 ) = 0 y ′ ( 0 ) = 0 y ′′ ( 0 ) = 1.32824 And step-size, h = 0.1 . Approach: Convert the given initial value problem into initial value problem for a system in normal form. Assume, x 1 ( t ) = y ( t ) x 2 ( t ) = y ′ ( t ) x 3 ( t ) = y ′′ ( t ) ( 1 ) Compare the normal form with- x ′ 1 ( t ) = f 1 ( t , x 1 , x 2 , … , x n ) , x ′ 2 ( t ) = f 2 ( t , x 1 , x 2 , … , x n ) , ⋮ x ′ n ( t ) = f n ( t , x 1 , x 2 , … , x n ) ( 2 ) To compute an approximation for the solution to the initial value problem- x ′ i = f i ( t , x 1 , … , x n ) x i ( t 0 ) = a i i = 1 , 2 , … , n for t 0 ≤ t ≤ c , use a programmed version of the vectorized classical fourth-order Runge-Kutta algorithm with step-size h , number of steps N - t = t + h x i = x i + 1 6 ( k i , 1 + 2 k i , 2 + 2 k i , 3 + k i , 4 ) i = 1 , . … , n n = 0 , 1 , 2 , … Here, h = c − t 0 N t = t 0 x 1 = a 1 , … , x n = a n For i = 1 , … , n , k i , 1 = h f i ( t , x 1 , … , x n ) k i , 2 = h f i ( t + h 2 , x 1 + 1 2 k 1 , 1 , … , x n + 1 2 k n , 1 ) k i , 3 = h f i ( t + h 2 , x 1 + 1 2 k 1 , 2 , … , x n + 1 2 k n , 2 ) k i , 4 = h f i ( t + h , x 1 + k 1 , 3 , … , x n + k n , 3 ) Calculation: Differentiate system of equations in ( 1 ) x ′ 1 ( t ) = x 2 ( t ) x ′ 2 ( t ) = x 3 ( t ) x ′ 3 ( t ) = y ′′′ ( t ) ( 3 ) Do substitution from equation ( 3 ) in the given initial value problem y ′′ ′ + y y ′′ = ( y ′ ) 2 − 1 y ( 0 ) = 0 y ′ ( 0 ) = 0 y ′′ ( 0 ) = 1

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

7th Edition

ISBN: 9780137394524

Author: R. Nagle, Edward Saff

Publisher: PEARSON+

See similar textbooks

Videos

Question

Chapter 5.3, Problem 27E

To determine

To approximate:

The solution to the given initial value problem and sketch the solution on the interval t∈[0,2].

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.3, Problem 26E

Chapter 5.3, Problem 28E

Students have asked these similar questions

14. Show that if a, b, and c are integers such that (a, b) = 1 and c | (a+b), then (c, a) = (c, b) = 1.

A retail chain is interested in determining whether a digital video point-of-purchase (POP) display would stimulate higher sales for a brand advertised compared to the standard cardboard point-of-purchase display. To test this, a one-shot static group design experiment was conducted over a four-week period in 100 different stores. Fifty stores were randomly assigned to the control treatment (standard display) and the other 50 stores were randomly assigned to the experimental treatment (digital display). Compare the sales of the control group (standard POP) to the experimental group (digital POP). What were the average sales for the standard POP display (control group)? What were the sales for the digital display (experimental group)? What is the (mean) difference in sales between the experimental group and control group? List the null hypothesis being tested. Do you reject or retain the null hypothesis based on the results of the independent t-test? Was the difference between the…

Using fixed point iteration and Newton Raphson methods to solve the following function: f(x) = e−0.5x(4-x)-2. Take xo-2 and n=5.

Chapter 5 Solutions

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

To approximate: The solution to the given initial value problem and sketch the solution on the interval t ∈ [ 0 , 2 ] .

Solution Summary: The author illustrates how to approximate the solution to the given initial value problem on the interval tin left[0,2right].

Videos

To approximate:

The solution to the given initial value problem and sketch the solution on the interval t∈[0,2].

Want to see the full answer?

Chapter 5 Solutions

To approximate: The solution to the given initial value problem and sketch the solution on the interval t ∈ [ 0 , 2 ] .

Solution Summary: The author illustrates how to approximate the solution to the given initial value problem on the interval tin left[0,2right].

Videos

To approximate: The solution to the given initial value problem and sketch the solution on the interval t∈[0,2].

Want to see the full answer?

Chapter 5 Solutions

To approximate:

The solution to the given initial value problem and sketch the solution on the interval t∈[0,2].