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Q5.3-22E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 1

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Oscillations and Nonlinear Equations. For the initial value problem $x'' + (0 . 1) (1 - x^{2}) x' + x = 0; x (0) = x_{o}, x' (0) = 0$ using the vectorized Runge–Kutta algorithm with h = 0.02 to illustrate that as t increases from 0 to 20, the solution x exhibits damped oscillations when $x_{o} = 1$ , whereas exhibits expanding oscillations when $x_{o} = 2.1,$ .

The result can get by the Runge-Kutta method.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Transform the equation

Here, the equation $x'' + (0 . 1) (1 - x^{2}) x' + x = 0 .$

The system can be written as:

$x_{1} = x (t) x_{2} = x' = x'_{1}$

The transform equation is:

role="math" localid="1664101777238" $x'_{1} = x_{2} x'_{2} = - x_{1} - 0 . 1 (1 - {x^{2}}_{1}) x_{2}$

The initial conditions are,

$x_{1} (0) = x (0) = x_{o} = 1, 2, 1 x_{2} (0) = x' (0) = 0$

Apply the Runge-Kutta method

Apply Matlab to find the results. And some results are;

T	For $x_{0} = 1$	For $x_{o} = 2.1$
0	1	2.1
0.02	0.9998	2.09957
0.04	0.9992	2.0983
0.06	0.9982	2.096
0.1	0.9950	2.0893
1	0.5441	1.0600
2	-0.36227	-0.9737

Applying the same procedure gets the result.

This is the required result.

Most popular questions for Math Textbooks

Find a general solution for the differential equation with x as the independent variable:

$y''' + 3 y'' - 4 y' - 6 y = 0$

In Problems 10–13, use the vectorized Euler method with h = 0.25 to find an approximation for the solution to the given initial value problem on the specified interval.

$y'' = t^{2} - y^{2}; y (0) = 0, y' (0) = 1 on [0, 1]$

In Problems 3-8, determine whether the given function is a solution to the given differential equation.

$y = \sin x + x^{2}$ , $\frac{d^{2} y}{{dx}^{2}} + y = x^{2} + 2$

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

$\frac{dx}{dt} + \cos x = \sin t, x (π) = 0$

Question: In Problems 3–8, determine whether the given function is a solution to the given differential equation.

$y = 3 \sin 2 x + e^{- x}$ , $y'' + 4 y = 5 e^{- x}$

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