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Chapter 9: Problem 5

For each initial value problem, calculate the Euler approximation for the solution on the interval [0,1] using \(n=4\) segments. Draw the graph of your approximation. (Carry out the calculations "by hand" with the aid of a calculator, rounding to two decimal places. Answers may differ slightly, depending on when you do the rounding.) $$ \begin{array}{l} y^{\prime}=3 x-2 y \\ y(0)=2 \end{array} $$

Short Answer

Expert verified
Using Euler's method, the approximate values are 1 at \(x = 0.25\), 0.69 at \(x = 0.5\), 0.72 at \(x = 0.75\), and 0.92 at \(x = 1\).

Step by step solution

01

Determine Step Size

The interval given is \[0, 1\] and it needs to be divided into \(n = 4\) equal segments. The step size \( h \) is calculated as \( h = \frac{1-0}{4} = 0.25 \).
02

Initialize Variables

Set the initial conditions with \( x_0 = 0 \) and \( y_0 = 2 \). These will be your starting values for the Euler method.
03

Apply Euler's Method for First Step

Calculate \( y_1 \) using Euler's formula: \[ y_{n+1} = y_n + h \, f(x_n, y_n) \]For the first step, \( f(x_0, y_0) = 3x_0 - 2y_0 = 3 \times 0 - 2 \times 2 = -4 \),\[ y_1 = 2 + 0.25 \times (-4) = 2 - 1 = 1 \].
04

Apply Euler's Method for Second Step

Now, calculate \( y_2 \) using the updated values:\( x_1 = 0.25 \), \( y_1 = 1 \),\[ f(x_1, y_1) = 3 \times 0.25 - 2 \times 1 = 0.75 - 2 = -1.25 \]\[ y_2 = 1 + 0.25 \times (-1.25) = 1 - 0.3125 = 0.69 \].
05

Apply Euler's Method for Third Step

Calculate \( y_3 \) using:\( x_2 = 0.5 \), \( y_2 = 0.69 \),\[ f(x_2, y_2) = 3 \times 0.5 - 2 \times 0.69 = 1.5 - 1.38 = 0.12 \]\[ y_3 = 0.69 + 0.25 \times 0.12 = 0.69 + 0.03 = 0.72 \].
06

Apply Euler's Method for Fourth Step

Finally, calculate \( y_4 \) using:\( x_3 = 0.75 \), \( y_3 = 0.72 \),\[ f(x_3, y_3) = 3 \times 0.75 - 2 \times 0.72 = 2.25 - 1.44 = 0.81 \]\[ y_4 = 0.72 + 0.25 \times 0.81 = 0.72 + 0.2025 = 0.92 \].
07

Graph the Approximation

Plot the points \((0, 2), (0.25, 1), (0.5, 0.69), (0.75, 0.72), (1, 0.92)\) on a graph, and connect them linearly to represent the Euler approximation on the interval [0,1].

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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 fundamental in expressing how quantities change in relation to each other. They are widely used across many scientific fields, including physics, biology, and engineering. A differential equation involves unknown functions and their derivatives.

In this context, consider the equation \( y' = 3x - 2y \) from the exercise. Here, \( y' \) represents the derivative of \( y \) with respect to \( x \). This specific type of differential equation is called a linear first-order differential equation.
  • The term \( 3x \) indicates how the rate of change of \( y \) depends linearly on \( x \).
  • The term \(-2y\) signifies how \( y \) regulates its own change.
These equations are expressed in a form \( \, y' = f(x, y) \, \), which shows how \( y \)'s rate of change is determined by both \( x \) and \( y \) itself. Understanding differential equations is crucial for modeling real-world phenomena where rates of change occur continuously.
Numerical Approximation
Numerical approximation is a method used to find approximate solutions to mathematical problems that might not have straightforward analytical solutions. This becomes useful in solving differential equations, especially when exact solutions are complex or impossible to determine.

The Euler Method is one such numerical technique. It's simple yet powerful, perfect for beginners to understand the concept of approximating solutions. Here鈥檚 briefly how it works:
  • You start with an initial known value, given by the problem.
  • You use tiny steps to estimate the value of the function at subsequent points based on its rate of change.
  • You repeat this process over the desired interval.
For example, in our exercise, we took step sizes of \( h = 0.25 \) to compute the successive values of \( y \) using the formula:\[y_{n+1} = y_n + h \, f(x_n, y_n)\]This iterative process gives us a sequence of approximations that converge to the function's behavior on the interval.
Initial Value Problem
An initial value problem in mathematics involves finding a function that satisfies a differential equation and fulfills an initial condition. Usually, you鈥檙e provided with the starting point or initial condition, which is necessary to uniquely determine the solution.

In our example, the initial value is given by \( y(0) = 2 \). This notation tells us where to begin our numerical approximation or analytical solution. By initiating the process from this point and applying a method like Euler's method, you generate an approximation of how \( y \) evolves over time from that starting point.
  • Initial values provide the anchor for solving differential equations.
  • They ensure the uniqueness of the solution by specifying a starting trend within the dynamic system.
Having a clear initial value makes it possible to employ methods like numerical approximation effectively, ensuring that you're nor just guessing, but rather following a set path through the solution space.

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Most popular questions from this chapter

The following problems extend and augment the material presented in the text. BIOMEDICAL: Fick's Law Fick's Law governs the diffusion of a solute across a cell membrane. According to Fick's Law, the concentration \(y(t)\) of the solute inside the cell at time \(t\) satisfies \(\frac{d y}{d t}=\frac{k A}{V}\left(C_{0}-y\right),\) where \(k\) is the diffusion constant, \(A\) is the area of the cell membrane, \(V\) is the volume of the cell, and \(C_{0}\) is the concentration outside the cell. a. Find the general solution of this differential equation. (Your solution will involve the constants \(k, A, V\) and \(C_{0}\).) b. Find the particular solution that satisfies the initial condition \(y(0)=y_{0},\) where \(y_{0}\) is the initial concentration inside the cell.

The water in a 100,000 -gallon reservoir contains 0.1 gram of pesticide per gallon. Each hour, 2000 gallons of water (containing 0.01 gram of pesticide per gallon) is added and mixed into the reservoir, and an equal volume of water is drained off. a. Write a differential equation and initial condition that describe the amount \(y(t)\) of pesticide in the reservoir after \(t\) hours. b. Solve this differential equation and initial condition. c. Graph your solution on a graphing calculator and find when the amount of pesticide will reach 0.02 gram per gallon, at which time the water is safe to drink. d. Use your solution to find the "long-run" amount of pesticide in the reservoir.

Explain why we cannot use an integrating factor \(I(x)\) to solve the nonlinear differential equation \(y^{\prime}+p(x) y^{2}=q(x) .\)

For each initial value problem: a. Use an Euler's method graphing calculator program to find the estimate for \(y(2)\). Use the interval [0,2] with \(n=50\) segments. b. Solve the differential equation and initial condition exactly by separating variables or using an integrating factor. c. Evaluate the solution that you found in part (b) at \(x=2\). Compare this actual value of \(y(2)\) with the estimate of \(y(2)\) that you found in part (a). $$ \begin{array}{l} y^{\prime}+y=2 e^{x} \\ y(0)=5 \end{array} $$

When you swallow a pill, the medication passes through your stomach lining into your bloodstream, where some is absorbed by the cells of your body and the rest continues to circulate for future absorption. The amount \(y(t)\) of medication remaining in the bloodstream after \(t\) hours can be modeled by the differential equation $$ \frac{d y}{d t}=a b e^{-b t}-c y $$ for constants \(a, b,\) and \(c\) (respectively the dosage of the pill, the dissolution constant of the pill, and the absorption constant of the medication). For the given values of the constants: a. Substitute the constants into the stated differential equation. b. Solve the differential equation (with the initial condition of having no medicine in the bloodstream at time \(t=0)\) to find a formula for the amount of medicine in the bloodstream at any time \(t\) (hours). c. Use your solution to find the amount of medicine in the bloodstream at time \(t=2\) hours. d. Graph your solution on a graphing calculator and find when the amount of medication in the bloodstream is maximized. \(a=10 \mathrm{mg}, \quad b=3, \quad c=0.2\)
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