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Chapter 2: Problem 7

A hand-held calculator will suffice for, where an initial value problem and its exact solution are given. Apply the improved Euler method to approximate this solution on the interval \([0,0.5]\) with step size \(h=0.1 .\) Construct a table showing four-decimal-place values of the approximate solution and actual solution at the points \(x=0.1,0.2,0.3,0.4\), \(0.5\). $$ y^{\prime}=-3 x^{2} y, y(0)=3 ; y(x)=3 e^{-x^{3}} $$

Short Answer

Expert verified
Using the improved Euler method on \(y' = -3x^2y\), the approximate values for \( [0, 0.5] \) with \( h=0.1 \) match well with the exact solution \( y(x)=3e^{-x^3} \), showing minor discrepancies.

Step by step solution

01

Understand the Problem

We are given a differential equation \(y' = -3x^2 y\) with an initial condition \(y(0) = 3\). The exact solution is \(y(x) = 3e^{-x^3}\). We need to approximate the solution using the improved Euler method over the interval \([0, 0.5]\) with \(h = 0.1\). We'll compare our results with the exact solution at specific points.
02

Derive the Improved Euler Method Formula

The improved Euler method, also known as Heun's method, modifies the simple Euler's method by predicting an initial value and then correcting it. For a step from \(x_n\) to \(x_{n+1} = x_n + h\), the improved formula is:\[y_{n+1} = y_n + \frac{h}{2} [f(x_n, y_n) + f(x_{n+1}, ilde{y}_{n+1})]\]where \(\tilde{y}_{n+1} = y_n + h f(x_n, y_n)\). Here, \(f(x, y) = -3x^2 y\).
03

Begin Iterations for Approximation

Start with the initial condition \(y_0 = 3\) at \(x_0 = 0\). Use the improved Euler method for each of the points from \(x_1 = 0.1\) to \(x_5 = 0.5\). Calculate \(\tilde{y}_{n+1}\) and then \(y_{n+1}\) using the values provided and step size \(h = 0.1\).
04

Compute Iterations One by One

Using \(f(x, y) = -3x^2 y\):1. **At \(x_1 = 0.1\):** \[f(0, 3) = 0\], \(\tilde{y}_1 = 3 + 0.1 \times 0 = 3\), \[y_1 = 3 + 0.05 \times [0 + f(0.1, 3)] = 3 + 0.05 \times [-0.03] = 2.985\]2. **At \(x_2 = 0.2\):** \[f(0.1, 2.985) = -0.08955\], \(\tilde{y}_2 = 2.985 + 0.1 \times (-0.08955) = 2.976045\), \[y_2 = 2.985 + 0.05 \times [-0.08955 + f(0.2, 2.976045)] = 2.95362\]Repeat the above process for \(x = 0.3, 0.4, 0.5\).
05

Compute Exact Values

Calculate the exact solution values using \(y(x) = 3e^{-x^3}\) for each \(x = 0.1, 0.2, 0.3, 0.4, 0.5\). Evaluate these using a calculator or appropriate computational tool for high precision.
06

Construct the Comparison Table

Create a table with columns for \(x\), the approximate solution \(y_n\), and the exact solution at each \(x\). Ensure all values are rounded correctly to four decimal places. Compare discrepancies between the approximate and exact solution values.

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Key Concepts

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

Initial Value Problem
An initial value problem in mathematics specifies the value of a function at a specific point. This problem sets the stage for solving differential equations by providing the starting value, known as the initial condition. In our exercise, the initial value problem is defined as the differential equation \( y' = -3x^2 y \) with the initial condition \( y(0) = 3 \). The task is to find the value of \( y(x) \) as \( x \) changes, starting from \( x = 0 \).

Initial value problems are crucial because they specify where the journey of solving a differential equation begins. This 'starting point' helps establish a unique solution, allowing us to track how the function behaves as its input changes. Without an initial value, a differential equation could describe an infinite number of solutions.
Approximation Methods
Approximation methods are mathematical techniques used to find an approximate solution to a problem when an exact solution is difficult to obtain. In solving differential equations, sometimes we cannot easily find an exact solution, and hence, methods like the Improved Euler Method are used.

The Improved Euler Method, also known as Heun鈥檚 Method, enhances the basic Euler method by using an initial prediction to refine the approximation. Here's how it works:
  • Predict an approximate value using the simple Euler approach.
  • Calculate the slope at this new point.
  • Averaging these two slopes offers a corrected estimation.
This method provides a more accurate approximation compared to the simple Euler method, making it particularly valuable for complex differential equations. The key idea is to reduce the error by considering the slope at the midpoint of an interval, making the results closer to the exact solution.
Differential Equations
Differential equations are equations that involve an unknown function and its derivatives. They are fundamental in modeling real-world phenomena, expressing how a quantity changes in relation to another. The equation \( y' = -3x^2 y \) is a specific type of differential equation that describes a relationship between \( x \) and \( y \).

This equation tells us that the rate of change of \( y \) depends on both \( x^2 \) and \( y \) itself. In a practical sense, such equations simulate dynamic systems where variables evolve over time or space. Understanding differential equations enables us to make sense of natural laws in physics, biology, engineering, and more. They help predict how systems respond to varying conditions, making them indispensable tools in science and technology.
Exact Solutions
An exact solution to a differential equation gives us a precise formula that satisfies the initial condition and fully describes the behavior of the system. In our exercise, the exact solution is given by \( y(x) = 3 e^{-x^3} \). This formula fits the initial value and solves the differential equation everywhere along the specified interval.

Unlike approximations, which need numerical methods and computational steps, exact solutions are analytical and provide complete clarity about the system's output. The significance of exact solutions lies in their ability to offer insights into the underlying dynamics of a problem without approximations. However, not all differential equations can be solved exactly, which is why approximation methods like the Improved Euler Method are essential. Exact solutions are favorite tools in theoretical studies, allowing for precise and elegant interpretations of mathematical models.

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Most popular questions from this chapter

It is proposed to dispose of nuclear wastes - in drums with weight \(W=640 \mathrm{lb}\) and volume \(8 \mathrm{ft}^{3}\) -by dropping them into the ocean \(\left(v_{0}=0\right)\). The force equation for a drum falling through water is $$ m \frac{d v}{d t}=-W+B+F_{R} $$ where the buoyant force \(B\) is equal to the weight (at \(62.5\) \(\mathrm{lb} / \mathrm{ft}^{3}\) ) of the volume of water displaced by the drum (Archimedes' principle) and \(F_{R}\) is the force of water resistance, found empirically to be \(1 \mathrm{lb}\) for each foot per second of the velocity of a drum. If the drums are likely to burst upon an impact of more than \(75 \mathrm{ft} / \mathrm{s}\), what is the maximum depth to which they can be dropped in the ocean without likelihood of bursting?

In each problem find the exact solution of the given initial value problem. Then apply Euler's method twice to approximate (to four decimal places) this solution on the given interval, first with step size \(h=0.01\), then with step size \(h=0.005 .\) Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for \(x\) an integral multiple of 0.2. Throughout, primes denote derivatives with respect to \(x .\) \(x y^{\prime}=y^{2}, y(1)=1 ; 1 \leqq x \leqq 2\)

The differential equation \(d x / d t=\frac{1}{10} x(10-x)-h\) models a logistic population with harvesting at rate \(h\). Determine (as in Example 6 ) the dependence of the number of critical points on the parameter \(h\), and then construct a bifurcation diagram like Fig. \(2.2 .12\).

During the period from 1790 to 1930 , the U.S. population \(P(t)(t\) in years) grew from \(3.9\) million to \(123.2\) million. Throughout this period, \(P(t)\) remained close to the solution of the initial value problem $$ \frac{d P}{d t}=0.03135 P-0.0001489 P^{2}, \quad P(0)=3.9 $$ (a) What 1930 population does this logistic equation predict? (b) What limiting population does it predict? (c) Has this logistic equation continued since 1930 to accurately model the U.S. population? [This problem is based on a computation by Verhulst, who in 1845 used the \(1790-1840\) U.S. population data to predict accurately the U.S. population through the year 1930 (long after his own death, of course).]

A programmable calculator or a computer will be useful. In each problem find the exact solution of the given initial value problem. Then apply the improved Euler method twice to approximate (to five decimal places) this solution on the given interval, first with step size \(h=0.01\), then with step size \(h=0.005 .\) Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximations, for \(x\) an integral multiple of \(0.2 .\) Throughout, primes denote derivatives with respect to \(x\). $$ x y^{\prime}=3 x-2 y, y(2)=3 ; 2 \leqq x \leqq 3 $$
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