Chapter 1: Q5.3-11E (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.
$(1 + t^{2}) y'' + y' - y = 0; y (0) = 1, y' (0) = - 1 on [0, 1]$

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

Short Answer

Step by step solution

Transform the equation

Apply Euler’s method.

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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.(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

Mechanics Maths

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Pure Maths

Statistics

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