Chapter 1: Q5.3-12E (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.
$t^{2} y'' + y = t + 2; y (1) = 1, y' (1) = - 1 on [1, 2]$

Short Answer

Expert verified

The solution is:

$y (1 . 25) = 0 . 75 y (1 . 5) = 0 . 625 y (1 . 75) = 0 . 6 y (2) = 0 . 6548$

Step by step solution

Transform the equation

Here h=0.25 on [0,2].

The equations can be written as:

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

The transformation of the equation is:

localid="1664089567642" $x'_{1} (t) = x_{2} (t) x'_{2} (t) = \frac{- x_{1} + t + 2}{t^{2}}$

The initial conditions are:

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

Apply Euler’s method

Now,

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

$t_{n + 1} = t_{n} + h t_{1} = 1 + 0.25 = 1.25 x_{1} (1 . 25) = x_{1, 1} = 0.75 x_{2} (1 . 25) = x_{2, 1} = - 0.5$

And

$t_{n + 1} = t_{n} + h t_{2} = 1.25 + 0.25 = 1.5 x_{1} (1 . 5) = x_{1, 2} = 0.625 x_{2} (1 . 5) = x_{2, 2} = - 0.1$

$t_{3} = 1.5 + 0.25 = 1.75 x_{1} (1 . 75) = x_{1, 3} = 0.6 x_{2} (1 . 75) = x_{2, 3} = - 0.219$

$t_{4} = 1.75 + 0.25 = 2 x_{1} (2) = x_{1, 4} = 0.6548 x_{2} (2) = x_{2, 4} = 0.4765$

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

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.t2y''+y=t+2;y(1)=1,y'(1)=-1on[1,2]

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

Applied Mathematics

Mechanics Maths

Geometry

Statistics

Logic and Functions

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