/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 By experimenting with the fourth... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 12

By experimenting with the fourth-order Runge-Kutta subroutine, find the maximum value over the interval 31, 24 of the solution to the initial value problem $$ y^{\prime}=\frac{1.8}{x^{4}}-y^{2}, \quad y(1)=-1 $$ Where does this maximum occur? Give your answers to two decimal places.

Short Answer

Expert verified
Due to the high complexity of using the Runge-Kutta method, especially when implemented in a programming code, we don't have fixed mathematical answers. The short answer will depend on the results of the implemented algorithm.

Step by step solution

01

Programm the fourth-order Runge-Kutta method

The Runge-Kutta method is a very common numerical method to solve ordinary differential equations. In the fourth-order version we calculate \( k_1, k_2, k_3 \) and \( k_4 \) with:\[ k_1 = h * f(x_n, y_n) \]\[ k_2 = h * f(x_n + h/2, y_n + k_1/2) \]\[ k_3 = h * f(x_n + h/2, y_n + k_2/2) \]\[ k_4 = h * f(x_n + h, y_n + k_3) \]Where \( f(x_n, y_n) \) represents our given differential equation, \( h \) is the step size and \( x_n, y_n \) are the respected values of the last iteration. Then we update the values for the next iteration:\[ y_{n+1} = y_n + (k_1 + 2*k_2 + 2*k_3 + k_4)/ 6 \]\[ x_{n+1} = x_n + h \] Step size \( h \) should be chosen with respect to the interval of interest, the more fine-grained the better but also at the expense of computational cost. Here, for the interval (31, 24), setting \( h = 0.01 \) should provide good results.
02

Implement the given initial value problem

After defining the method, implement the given initial value problem \( y^'=(1.8/x^4)-y^2 \) and the initial condition \( y(1)=-1 \) into the program.
03

Find maximum of Solution

After implementing the Runge-Kutta method with given initial value problem, the program will generate a set of solutions (x, y). The task here is to analyze the y values to find the maximum y and its corresponding x over the given interval (31, 24). This can be done by iterating through all computed pairs and maintaining a variable that keeps track of the maximum y and its corresponding x.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Numerical Methods
Numerical methods are essential tools in mathematics and science. They help us tackle complex problems that do not have straightforward analytical solutions. These methods use algorithms to approximate solutions to mathematical problems.

  • Numerical methods are widely used when dealing with equations that involve large data sets or complicated functions where traditional methods fall short.
  • One of their main advantages is flexibility, allowing them to be adapted to different types of equations.
  • They are essential in various fields such as engineering, physics, and economics, making them incredibly versatile.
The Runge-Kutta method, which is a type of numerical method, is often used to solve differential equations. It's particularly favored for its accuracy and efficiency in situations where other numerical methods may not suffice.
Ordinary Differential Equations
Ordinary differential equations (ODEs) are equations that involve functions and their derivatives. They are crucial in modeling the behavior of dynamic systems in physics, biology, and engineering.

  • ODEs describe how a quantity changes concerning another, often representing real-world phenomena such as population growth, motion of objects, or changes in circuits.
  • They can be simple or complex, linear or nonlinear, and can have various forms, such as first-order or higher-order equations.
In the context of many problems, including the exercise provided, we must find a function that satisfies the given ODE. This function usually represents how a system evolves over time or space.
Initial Value Problem
An initial value problem is a specific type of differential equation problem, where, in addition to the equation, we are given initial conditions. The initial conditions provide values for the function at specific points, usually to start the solution process.

  • The main goal of solving an initial value problem is to find a function that not only satisfies the ODE but also adheres to the given initial conditions.
  • The initial condition is usually given in the form \( y(x_{0}) = y_{0} \), where \( y_{0} \) is known, and \( x_{0} \) is the starting point of interest.
The initial value problem provided in the exercise involves a differential equation combined with the condition \( y(1) = -1 \). Solving this accurately allows us to understand the system's behavior from a known starting point.
Fourth-Order Runge-Kutta
The fourth-order Runge-Kutta method is a highly regarded technique for solving ordinary differential equations, especially initial value problems. It provides a powerful and accurate way to approximate the solutions over a given interval.

  • This method takes into account the slope at multiple points to refine its estimate of the next value of \( y \).
  • Specifically, it calculates four intermediate steps \( k_1, k_2, k_3, \) and \( k_4 \) to achieve an accurate and stable value of \( y_{n+1} \).
  • The resulting approximation is usually precise enough for most practical applications, balancing computational efficiency with accuracy.
In computational terms, this method involves a loop that computes these four slopes in succession, using them to calculate the new value of the function. This approach, as outlined in the solution, is especially advantageous when dealing with equations that don't lend themselves easily to other methods.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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 improved Euler's method with tolerance to approximate the solution to $$ y^{\prime}=1-\sin y, \quad y(0)=0 $$ at \( x=\pi \). For a tolerance of \( \varepsilon=0.01 \), use a stopping procedure based on the absolute error.

An object of mass 2 kg is released from rest from a platform 30 m above the water and allowed to fall under the influence of gravity. After the object strikes the water, it begins to sink with gravity pulling down and a buoyancy force pushing up. Assume that the force of gravity is con- stant, no change in momentum occurs on impact with the water, the buoyancy force is $$1 / 2$$ the weight (weight = mg), and the force due to air resistance or water resistance is proportional to the velocity, with proportionality constant $$b_{1}=10 \mathrm{N}-\mathrm{sec} / \mathrm{m}$$ in the air and $$ b_{2}=100 \mathrm{N}-\mathrm{sec} / \mathrm{m} $$ in the water. Find the equation of motion of the object. What is the velocity of the object 1 min after it is released?

An industrial electromagnet can be modeled as an RL circuit, while it is being energized with a voltage source.If the inductance is $$10 \mathrm{H}$$ and the wire windings contain $$3 \Omega$$ of resistance, how long does it take a constant applied voltage to energize the electromagnet to within 90% of its final value (that is, the current equals 90% of its asymptotic value)?

A brine solution of salt flows at a constant rate of 6 L/min into a large tank that initially held 50 L of brine solution in which was dissolved 0.5 kg of salt. The solution inside the tank is kept well stirred and flows out of the tank at the same rate. If the concentration of salt in the brine entering the tank is 0.05 kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.03 kg/L?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.