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Chapter 2: Problem 12

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 \(x .\) \(y^{\prime}=\frac{1}{2}(y-1)^{2}, y(0)=\overline{2} ; 0 \leqq x \leqq 1\)

Short Answer

Expert verified
Exact solution: \( y(x) = 1 + \frac{1}{1 - \frac{x}{2}} \); compare Euler's approximations with exact values and note errors.

Step by step solution

01

Solve the Differential Equation

The differential equation is given by \( y' = \frac{1}{2}(y-1)^2 \) with the initial condition \( y(0) = 2 \). This is a separable differential equation. To solve it, separate variables:\[ \frac{dy}{(y-1)^2} = \frac{1}{2} dx \]Integrating both sides, we have:\[ \int \frac{dy}{(y-1)^2} = \int \frac{1}{2} dx \]The left side integrates to \( -\frac{1}{y-1} \) and the right side to \( \frac{1}{2}x + C \).Thus, \( -\frac{1}{y-1} = \frac{1}{2}x + C \).Using the initial condition \( y(0) = 2 \):\[ -\frac{1}{2-1} = \frac{1}{2}(0) + C \Rightarrow C = -1 \]So, \( -\frac{1}{y-1} = \frac{1}{2}x - 1 \), which simplifies to:\[ y = 1 \pm \frac{1}{1 - \frac{x}{2}} \]Using the initial condition \( y(0)=2 \), we determine that the positive solution of the \( \pm \) is valid, leading to:\( y(x) = 1 + \frac{1}{1 - \frac{x}{2}} \).
02

Apply Euler's Method with h=0.01

Euler's method approximates solutions to differential equations by iteratingthe equation: \( y_{n+1} = y_n + h f(x_n, y_n) \).Starting with \( y(0)=2 \), calculate subsequent \( y \)-values on the interval \( 0 \leq x \leq 1 \) with a step size \( h = 0.01 \).The general term for our specific equation is:\[ y_{n+1} = y_n + 0.01 \times \frac{1}{2}(y_n - 1)^2 \]Start with \( y_0 = 2 \) and iterate through \( x = 0.01, 0.02, \ldots, 1 \).Compute these values to complete Euler's approximation table for \( h=0.01 \).
03

Apply Euler's Method with h=0.005

Similarly, use Euler's method with a smaller step size \( h = 0.005 \).The update equation remains:\[ y_{n+1} = y_n + 0.005 \times \frac{1}{2}(y_n - 1)^2 \]Again, begin with \( y_0 = 2 \) and iterate through \( x = 0.005, 0.01, 0.015, \ldots, 1 \).This will yield a more accurate approximation of \( y(x) \).
04

Compare with Exact Solution and Calculate Percentage Error

For each integral multiple of 0.2 (i.e., at \( x = 0.2, 0.4, 0.6, 0.8, \) and \( 1.0 \)), compare the approximate values from the Euler method (using \( h = 0.005 \) for more accuracy) with the exact solution:\[ y(x) = 1 + \frac{1}{1 - \frac{x}{2}} \]Calculate the percentage error using the formula:\[ \text{Percentage Error} = \left| \frac{\text{Approximate Value} - \text{Exact Value}}{\text{Exact Value}} \right| \times 100\% \]Fill in a table with these comparisons and calculated percentage errors.

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Key Concepts

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

Differential Equations
Differential equations are equations that involve a function and its derivatives. They model how things change and are essential in fields like physics, engineering, and biology.
In this problem, we're dealing with a first-order differential equation, written as:\[ y' = \frac{1}{2}(y-1)^2 \]This equation relates the rate of change of \(y\) to its current state. The task is to find a function \(y(x)\) that satisfies this relationship, starting from a known initial condition, \(y(0) = 2\).
Differential equations can be tricky because they involve unknown functions. However, they offer powerful tools for modeling real systems. By finding a solution, you can predict how a system evolves over time!
Euler's Method
Euler's method is a simple numerical tool used to approximate the solutions of differential equations. It works by using a step-by-step approach to estimate the curve described by the differential equation.
The core idea is straightforward. Starting from an initial condition, you move in small steps, using the slope given by the differential equation to find the next value.
For our problem, Euler's method uses the update formula:
  • \( y_{n+1} = y_n + h \cdot f(x_n, y_n) \)
  • Where \( f(x, y) = \frac{1}{2}(y - 1)^2 \)
You iterate this process, adjusting the step size, \(h\), to improve accuracy. In our example, we try \(h = 0.01\) and \(h = 0.005\), illustrating how smaller steps can lead to more precise approximations.
Separable Equations
Separable differential equations are a special type where you can separate the variables to opposite sides of the equation, making them easier to solve.
For our equation, \( y' = \frac{1}{2}(y-1)^2 \), we can re-arrange it to:\[ \frac{dy}{(y-1)^2} = \frac{1}{2} dx \]This separation of variables allows us to integrate each side independently.
  • Integrate \( \int \frac{dy}{(y-1)^2} \) to find \(-\frac{1}{y-1}\).
  • Integrate \( \int \frac{1}{2} dx \) to find \( \frac{1}{2}x + C \).
With the initial condition \( y(0) = 2 \), we can solve for the constant \(C\), making it possible to find the exact solution of the differential equation.
Separable equations are often more manageable and provide a clear path from the differential form to a functional solution.
Approximation Accuracy
Approximation accuracy is crucial when using numerical methods like Euler's method. The smaller the step size, the closer the approximation aligns with the exact solution.
In this exercise, we compute approximations using two step sizes, \( h = 0.01 \) and \( h = 0.005 \). Smaller step sizes, while more computationally intensive, decrease the error in our approximation.
Once the approximate values are calculated, we evaluate their accuracy by comparing them to the exact solution:\[ y(x) = 1 + \frac{1}{1 - \frac{x}{2}} \]
To measure how far off our approximate values are, we calculate the percentage error using:
  • \( \text{Percentage Error} = \left| \frac{\text{Approximate Value} - \text{Exact Value}}{\text{Exact Value}} \right| \times 100\% \)
This percentage helps us understand the precision of our approximation鈥攁n important factor when deciding which numerical method or step size suits our needs.

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Most popular questions from this chapter

A hand-held calculator will suffice for Problems 1 through 10, where an initial value problem and its exact solution are given. Apply the Runge-Kutta method to approximate this solution on the interval \([0,0.5]\) with step size \(h=0.25 .\) Construct a table showing five-decimal-place values of the approximate solution and actual solution at the points \(x=0.25\) and \(0.5\). $$ y^{\prime}=-y, y(0)=2 ; y(x)=2 e^{-x} $$

(a) Show that if a projectile is launched straight upward from the surface of the earth with initial velocity \(v_{0}\) less than escape velocity \(\sqrt{2 G M} / R\), then the maximum distance from the center of the earth attained by the projectile is $$ r_{\mathrm{max}}=\frac{2 G M R}{2 G M-R v_{0}^{2}} $$ where \(M\) and \(R\) are the mass and radius of the earth, respectively. (b) With what initial velocity \(v_{0}\) must such a projectile be launched to yield a maximum altitude of 100 kilometers above the surface of the earth? (c) Find the maximum distance from the center of the earth, expressed in terms of earth radii, attained by a projectile launched from the surface of the earth with \(90 \%\) of escape velocity.

A motorboat starts from rest (initial velocity \(v(0)=v_{0}=\) 0). Its motor provides a constant acceleration of \(4 \mathrm{ft} / \mathrm{s}^{2}\) but water resistance causes a deceleration of \(v^{2} / 400 \mathrm{ft} / \mathrm{s}^{2}\) Find \(v\) when \(t=10 \mathrm{~s}\), and also find the limiting velocity as \(t \rightarrow+\infty\) (that is, the maximum possible speed of the boat).

An arrow is shot straight upward from the ground with an initial velocity of \(160 \mathrm{ft} / \mathrm{s}\). It experiences both the deceleration of gravity and deceleration \(v^{2} / 800\) due to air resistance. How high in the air does it go?

Use either the exact solution or a computer-generated slope field to sketch the graphs of several solutions of the given differential equation, and highlight the indicated particular solution. $$ \frac{d x}{d t}=9-4 x^{2}, x(0)=0 $$
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