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Q10E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 259

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

In Problems 10–13, use the vectorized Euler method with = 0.25 to find an approximation for the solution to the given initial value problem on the specified interval.
$y'' + ty' + y = 0; y (0) = 1, y' (0) = 0 on [0, 1]$

$y (0 . 25) = 1 y (0 . 5) = 0 . 9375 y (0 . 75) = 0 . 8164 y (1) = 0 . 651855$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Transform equation

Here h=0.25 0n [0,1]

The equations can be written as;

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

The transformation of the equation is;

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

The initial conditions are;

$x_{1} (0) = y_{1} (0) = 1 = x_{1, 0} x_{2} (0) = y' (0) = 0 = x_{2, 0}$

Step 2: Apply Euler’s method.

Now,

$x_{n + 1} = x_{n} + hf (t_{n}, x_{n})$

$t_{n + 1} = t_{n} + h = 0 + 0.25 x_{1} (0 . 25) = x_{1, 1} = 1 x_{2} (0 . 25) = x_{2, 1} = - 0.25$

And

$t_{n + 1} = t_{n} + h t_{2} = 0.25 + 0.25 x_{1} (0 . 5) = x_{1, 2} = 0.9375 x_{2} (0 . 5) = x_{2, 2} = - 0.484375$

$t_{3} = 0.5 + 0.25 = 0.75 x_{1} (0 . 75) = x_{1, 3} = 0.816406 x_{2} (0 . 75) = x_{2, 3} = - 0.658203$

$t_{4} = 0.75 + 0.25 = 1 x_{1} (1) = x_{1, 4} = 0.651855 x_{2} (1) = x_{2, 4} = - 0.738891$

This is the required result.

Most popular questions for Math Textbooks

Using the software, sketch the direction field in the phase-plane for the system $\frac{d x}{d t} = y, \frac{d y}{d t} = - x - x^{3} .$ From the sketch, conjecture whether all solutions of this system are bounded. Solve the related phase plane differential equation and confirm your conjecture.

Find all the critical points of the system

$\frac{d x}{d t} = x^{2} - 1 \frac{d y}{d t} = x y$

and the solution curves for the related phase plane differential equation. Thereby proving that two trajectories lie on semicircles. What are the endpoints of the semicircles?

In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system.

$\frac{d x}{d t} = x^{2} - 2 y^{- 3}, \frac{d y}{d t} = 3 x^{2} - 2 x y$

In Problems 23 and 24, show that the given linear system is degenerate. In attempting to solve the system, determine whether it has no solutions or infinitely many solutions.

$(D - 1) [x] + (D - 1) [y] = - 3 e^{- 2 t}, (D + 2) [x] + (D + 2) [y] = 3 e^{t}$

In Problems 3 – 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

$x' + y' - x = 5, x' + y' + y = 1$

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