Chapter 1: Q5.3-13E (page 1)

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

Short Answer

Expert verified

The required solution is;

$y (0 . 25) = 0 . 25 y (0 . 5) = 0 . 5 y (0 . 75) = 0 . 75 y (1) = 1$

Step by step solution

Transform the equation

Here h=0.25 on [0,1].

The equations can be written as;

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

The transformation of the equation is;

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

The initial conditions are;

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

Apply Euler’s method

Now,

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

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

And

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

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

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

This is the required result.

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

Short Answer

Step by step solution

Transform the equation

Apply Euler’s method

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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''=t2-y2;y(0)=0,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

Calculus

Statistics

Applied Mathematics

Probability and Statistics

Mechanics Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

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