Problem 62 - Calculus Single Variable

Question

An initial value problem is given, along with its exact solution. (Read the instructions for Exercises \(47-50\) for terminology.) Verify that the given solution is correct by substituting it into the given differential equation and the initial value condition. Calculate the Euler's Method approximation \(y_{1}=y_{0}+F\left(x_{0}, y_{0}\right) \Delta x\) of \(y\left(x_{1}\right)\) where \(\Delta x=x_{1}-x_{0} .\) Let \(m_{1}=\left(F\left(x_{0}, y_{0}\right)+F\left(x_{1}, y_{1}\right)\right) / 2\) and \(z_{1}=y_{0}+\) \(m_{1} \Delta x .\) This is the Improved Euler Method approximation of \(y\left(x_{1}\right) .\) Calculate \(z_{1} .\) By evaluating \(y\left(x_{1}\right),\) determine which of the two approximations, \(y_{1}\) or \(z_{1},\) is more accurate. $$ \begin{array}{l} d y / d x=1+y / x, y(2)=1 / 2, x_{1}=3 / 2 ; \text { Exact solution: } y(x)= \\\ x \ln (x)+x(1 / 4-\ln (2)) \end{array} $$

Step-by-Step Solution

Answer

The Improved Euler's approximation \( z_1 = 5/96 \) is more accurate than the Euler's approximation \( y_1 = 1/8 \) for \( y(3/2) \).

Step 1: Verify the Exact Solution

First, substitute the exact solution \( y(x) = x \ln(x) + x(1/4 - \ln(2)) \) into the given differential equation \( \frac{dy}{dx} = 1 + \frac{y}{x} \). Calculate the derivative \( \frac{dy}{dx} \) using the chain rule and product rule. Verify that substituting \( y(x) \) and \( \frac{dy}{dx} \) into the differential equation satisfies it, confirming the solution is correct.

Step 2: Check Initial Condition

Substitute \( y(x) = x \ln(x) + x(1/4 - \ln(2)) \) with \( x = 2 \) into the solution to get \( y(2) = 1/2 \). Since \( y(2) = 1/2 \) matches the given initial condition exactly, this confirms the initial condition is satisfied.

Step 3: Calculate Euler's Method Approximation

To find the Euler's Method approximation \( y_1 \), use \( y_0 = 1/2 \), \( x_0 = 2 \), and \( x_1 = 3/2 \) with \( F(x_0, y_0) = 1 + y_0/x_0 = 1 + 1/4 = 5/4 \). Compute \( \Delta x = x_1 - x_0 = -1/2 \) and then \( y_1 = y_0 + F(x_0, y_0) \Delta x = 1/2 + (5/4)(-1/2) = 1/8 \).

Step 4: Calculate Improved Euler Method Approximation

First compute \( y_1 = 1/8 \) as in Euler's method. Evaluate \( F(x_1, y_1) = 1 + \frac{1/8}{3/2} = 35/24 \). Then, calculate \( m_1 = \left(F(x_0, y_0) + F(x_1, y_1)\right) / 2 = \left(5/4 + 35/24\right) / 2 = 55/48 \). Finally, find \( z_1 = y_0 + m_1 \Delta x = 1/2 + (55/48)(-1/2) = 5/96 \).

Step 5: Evaluate and Compare with Exact Solution

Substitute \( x = 3/2 \) into the exact solution \( y(x) = x \ln(x) + x(1/4 - \ln(2)) \) to find \( y(3/2) = 3/2 \ln(3/2) + 3/2(1/4 - \ln(2)) \). Compute this value to compare against \( y_1 = 1/8 \) and \( z_1 = 5/96 \). Determine the closer approximation to \( y(3/2) \).

Key Concepts

Euler's Method

Euler's Method is a straightforward numerical technique for estimating solutions to differential equations. It's frequently used when an exact solution is difficult to achieve, making it essential in fields like physics and engineering.
To apply Euler's Method, begin with an initial condition. This method estimates the value of a function at a future point by advancing step-by-step over small intervals, denoted as \( \Delta x \). The method uses the formula:

• \( y_{1}=y_{0}+F(x_{0}, y_{0}) \Delta x \)

With this formula, we iterate the process to approximate solutions across the desired domain. The method's simplicity makes it ideal for gaining initial insights, but its rough estimates mean that results might lack precision over larger intervals.

Improved Euler Method

The Improved Euler Method, sometimes known as the Heun's method or modified Euler method, enhances the basic Euler's Method by reducing errors in approximation. While Euler's Method offers a simple estimate, its sequential nature can lead to large discrepancies in solving differential equations. This is particularly true over larger intervals.
The Improved Euler Method tackles this by integrating additional information about the slope at the beginning and end of each interval. Specifically, it averages the slope \( F(x_0, y_0) \) and the predicted slope at the next step \( F(x_1, y_1) \) to better estimate the function's future value. The formula looks like this:

• \( m_{1}=\left(F(x_{0}, y_{0})+F(x_{1}, y_{1})\right) / 2 \)
• \( z_{1}=y_{0}+ m_{1} \Delta x \)

This approach corrects the endpoint by considering both the initial and estimated final slope, leading to a more accurate approximation.

Differential Equations

Differential equations are equations involving the derivatives of a function. They are pivotal in various scientific fields as they model the behavior of dynamic systems. Understanding them is key to physics, engineering, and other sciences to represent real-world processes mathematically.
In an initial value problem, a specific solution of a differential equation is required, typically starting from an initial condition. For example, given an equation \( dy/dx = 1 + y/x \) with an initial condition \( y(2) = 1/2 \), you aim to determine the function \( y(x) \) that satisfies these conditions. These types of problems necessitate approaches like Euler's or Improved Euler's Method to find approximate solutions when exact ones are infeasible.
The primary goal is to understand how the solution evolves from the given initial condition and behaves across its domain.

Exact Solution Verification

Verification of the exact solution of a differential equation is a critical step. This process confirms whether your proposed solution accurately satisfies both the differential equation and any initial conditions. To verify, substitute the proposed solution back into the original equation and check for consistency.
As illustrated in the problem, you might have a proposed solution like \( y(x) = x \ln(x) + x(1/4 - \ln(2)) \). By differentiating and substituting back into the given \( dy/dx = 1+y/x \), you should confirm this solution satisfies both the differential equation and conditions such as \( y(2) = 1/2 \). If both hold, your verification process succeeds.
These checks ensure the mathematical model you've developed reflects reality, crucial for future predictions and analysis.