Chapter 5: Q10E (page 259)

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.
$y'' + ty' + y = 0; y (0) = 1, y' (0) = 0 on [0, 1]$

Short Answer

Expert verified

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

Step by step solution

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;

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

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.

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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.
$y'' + ty' + y = 0; y (0) = 1, y' (0) = 0 on [0, 1]$

Short Answer

Step by step solution

Transform equation

Apply Euler’s method.

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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)=0on[0,1]

Short Answer

Step by step solution

Transform equation

Apply Euler’s method.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Applied Mathematics

Probability and Statistics

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

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.
$y'' + ty' + y = 0; y (0) = 1, y' (0) = 0 on [0, 1]$