Q5.3-11E In Problems 10鈥�13, use the vec... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 1: Q5.3-11E (page 1)

In Problems 10鈥�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.
$(1 + t^{2}) y'' + y' - y = 0; y (0) = 1, y' (0) = - 1 on [0, 1]$

Short Answer

Expert verified

The solution is:

$\begin{array}{rcl} y (0 . 25) = 0 . 75 \\ y (0 . 5) = 0 . 625 \\ y (0 . 75) = 0 . 57353 \\ y (1) = 0 . 563603 \end{array}$

Step by step solution

Transform the 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 theequationis;

localid="1664086646226" $x'_{1} (t) = x_{2} (t) x'_{2} (t) = \frac{1}{1 + t^{2}} x_{1} - \frac{1}{1 + t^{2}} x_{2}$

The initial conditions are;

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

Apply Euler’s method.

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

$t_{n + 1} = t_{n} + h t_{1} = 0 + 0.25 x_{1} (0 . 25) = x_{1, 1} = 0.75 x_{2} (0 . 25) = x_{2, 1} = - 0.5$

And

$t_{n + 1} = t_{n} + h t_{2} = 0.25 + 0.25 = 0.5 x_{1} (0 . 5) = x_{1, 2} = 0.625 x_{2} (0 . 5) = x_{2, 2} = - 0.20588$

$t_{3} = 0.5 + 0.25 = 0.75 x_{1} (0 . 75) = x_{1, 3} = 0.57353 x_{2} (0 . 75) = x_{2, 3} = - 0.039704$

role="math" localid="1664086890125" $t_{4} = 0.75 + 0.25 = 1 x_{1} (1) = x_{1, 4} = 0.563604 x_{2} (1) = x_{2, 4} = 0.058413$

This is the required result.

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91影视

In Problems 10鈥�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.
$(1 + t^{2}) y'' + y' - y = 0; y (0) = 1, y' (0) = - 1 on [0, 1]$

Short Answer

Step by step solution

Transform the equation

Apply Euler’s method.

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91影视

In Problems 10鈥�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.(1+t2)y''+y'-y=0;y(0)=1,y'(0)=-1on[0,1]

Short Answer

Step by step solution

Transform the equation

Apply Euler’s method.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Pure Maths

Logic and Functions

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 10鈥�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.
$(1 + t^{2}) y'' + y' - y = 0; y (0) = 1, y' (0) = - 1 on [0, 1]$