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Chapter 3: Problem 10

Use the improved Euler's method subroutine with step size \( h=0.1 \) to approximate the solution to $$ y^{\prime}=4 \cos (x+y), \quad y(0)=1 $$ at the points \( x=0,0.1,0.2, \dots, 1.0 \) Use your answers to make a rough sketch of the solution on [0, 1].

Short Answer

Expert verified
The solution to this exercise involves implementing the Improved Euler's method for the given ordinary differential equation and step size, calculating the numerical solution at points \(x_0, x_1, ..., x_{10}\), and then plotting the solution on the interval [0, 1]. The exact visual representation of the graph depends on the calculated numerical solutions.

Step by step solution

01

Initialize values

We initialize the required values: initial point a = 0, final point b = 1, step size h = 0.1, initial y value \(y_0\) = 1, and the given differential equation \(f(x, y) = 4 cos (x + y)\).
02

Implement Improved Euler's Method

Now implement the Improved Euler's method. This iterative technique includes two stages. The first stage generates an intermediate value, and the second stage improves that intermediate value. Use the following formulas for each point \(x_i\) in the range [a, b] with \(x_i = a + i * h\): First stage: \(k_1 = h * f(x_i, y_i)\), Second stage: \(k_2 = h * f(x_i + h, y_i + k_1)\), Update: \(y_{i+1} = y_i + 0.5 * (k_1 + k_2)\). Repeat until \(x_i\) reaches b.
03

Calculate solutions

Run the Improved Euler's Method and calculate all approximate solutions \(y_i\) to the differential equation at points \(x_i\) for \(i = 0, 1, 2, ..., 10\).
04

Plot the results

Use the calculated values of \(y_i\) at each \(x_i\) to make a rough sketch of the solution on the interval [0, 1]. The x-axis should represent 'x' values and the y-axis should represent 'y' values of the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Improved Euler's Method
The Improved Euler's Method is a numerical technique used to find approximate solutions to differential equations. It is an enhancement of the basic Euler's method and involves two stages to improve the accuracy of the solution. In the first stage, an intermediate value is calculated, and in the second stage, this value is refined to produce a more accurate result. The method follows these steps:
  • An initial calculation, using a given formula for a differential equation, provides a preliminary result.
  • A second calculation is done using this preliminary value, which produces a corrected step.
  • The final result is an average of these two values.
Each iteration refines the solution, making this method both easy to use and more precise than the basic Euler's approach. Using this method requires defining an initial condition, a step size, and iterating over a specified interval.
Differential Equations
Differential equations are mathematical expressions that describe how a function changes. They involve derivatives, which are measurements of how a quantity changes as its input changes. Differential equations can describe a wide range of real-world phenomena, from simple motion to complex dynamic systems.
There are different types of differential equations, but the one used in this problem is an ordinary differential equation (ODE), involving only one independent variable and its derivative.
For any given differential equation, a solution describes how the dependent variable changes with the independent variable. Solving these equations analytically can be complex, so numerical methods, like the Improved Euler's, offer a practical way to find approximate solutions.
Step-by-Step Solution
A step-by-step solution is crucial for understanding how numerical methods are applied to solve differential equations. It breaks down the process into manageable parts, allowing a learner to follow each calculation and see how the final solution is built.
For Improved Euler's Method:
  • Step 1: Initialize all given values, including the initial point, final point, step size, and initial condition.
  • Step 2: Perform calculations in two stages for each point in the interval. First, calculate the intermediate step. Next, refine it to improve accuracy.
  • Step 3: Use the results to update the next value in the sequence. This involves computing parameters like k1 and k2 to find the next y value.
  • Step 4: Continue iterating this process until you reach the final point.
Following this structured procedure, students can solve complex equations in parts, making it less intimidating and easy to replicate in various problems.
Initial Value Problem
An initial value problem (IVP) in differential equations involves finding a solution that satisfies a given initial condition. This initial condition specifies the value of the function at a particular point, usually at the beginning of the interval.
For the exercise provided, the initial value is given by y(0) = 1. This means that at time x = 0, the value of y must start at 1. The goal is then to predict how y changes over time.
IVPs are common in real-life applications because they embody the idea of starting conditions affecting the evolution of a system, such as the initial position and speed of a moving object determining its future trajectory. Solving IVPs often requires numerical techniques, like the Improved Euler's Method, to find approximate solutions at various points.

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Most popular questions from this chapter

A rotating flywheel is being turned by a motor that exerts a constant torque $$T$$ (see Figure 3.10). A retarding torque due to friction is proportional to the angular velocity $$\omega$$. If the moment of inertia of the flywheel is $$l$$ and its initial angular velocity is $$\omega_{0 h}$$ find the equation for the angular velocity $$\omega$$ as a function of time. [Hint: Use Newton's second law for rotational motion, that is, moment of inertia * angular acceleration = sum of the torques.]

Escape Velocity. According to Newton's law of gravitation, the attractive force between two objects varies inversely as the square of the distances between them. That is, $$F_{g}=G M_{1} M_{2} / r^{2}$$, where $$M_{1}$$ and $$M_{2}$$ are the masses of the objects, $$r$$ is the distance between them (center to center). $$F_{g}$$ is the attractive force, and $$G$$ is the constant of proportionality. Consider a projectile of constant mass $$m$$ being fired vertically from Earth (see Figure 3.12). Let $$t$$ represent time and $$v$$ the velocity of the projectile. (a) Show that the motion of the projectile, under Earth's gravitational force. is governed by the equation $$\frac{d v}{d t}=-\frac{g R^{2}}{r^{2}}$$ where $$r$$ is the distance between the projectile and the center of Earth, $$R$$ is the radius of Earth, $$M$$ is the mass of Earth, and $$g=G M / R^{2}$$. (b) Use the fact that $$d r / d t=v$$ to obtain $$v \frac{d v}{d r}=-\frac{g R^{2}}{r^{2}}$$ (c) If the projectile leaves Earth's surface with velocity $$v_{0}$$, show that $$v^{2}=\frac{2 g R^{2}}{r}+v_{0}^{2}-2 g R$$ (d) Use the result of part (c) to show that the velocity of the projectile remains positive if and only if $$v_{0}^{2}-2 g R>0$$. The velocity $$v_{e}=\sqrt{2 g R}$$ is called the escape velocity of Earth. (e) If $$g=9.81 \mathrm{m} / \mathrm{sec}^{2}$$ and $$R=6370 \mathrm{km}$$ for Earth, what is Earth's escape velocity? (f) If the acceleration due to gravity for the moon is $$g_{m}=g / 6$$ and the radius of the moon is $$R_{m}= 1738 km$$, what is the escape velocity of the moon?

The solution to the initial value problem $$ \frac{d y}{d x}+\frac{y}{x}=x^{3} y^{2}, \quad y(1)=3 $$ has a vertical asymptote ("blows up") at some point in the interval 31, 24 . By experimenting with the improved Euler's method subroutine, determine this point to two decimal places.

Use the fourth-order Runge-Kutta algorithm to approximate the solution to the initial value problem $$ y^{\prime}=1-x y, \quad y(1)=1 $$ at x = 2. For a tolerance of e = 0.001, use a stopping procedure based on the absolute error.

The solution to the initial value problem $$ \frac{d y}{d x}=y^{2}-2 e^{x} y+e^{2 x}+e^{x}, \quad y(0)=3 $$ has a vertical asymptote ("blows up") at some point in the interval 30, 24 . By experimenting with the fourth-order Runge-Kutta subroutine, determine this point to two decimal places.
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