Problem 31 - 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. $$ \begin{array}{l} d y / d x=x^{2}-2 y, y(0)=3, x_{1}=1 / 4 ; \text { Exact solution: } y(x)= \\ x^{2} / 2-x / 2+1 / 4+11 / 4 \exp (-2 x) \end{array} $$

Step-by-Step Solution

Answer

The Improved Euler Method approximation \(z_1\) is more accurate than \(y_1\).

Step 1: Verify the Exact Solution

Start by substituting the exact solution into the differential equation and the initial condition. The differential equation is \(\frac{dy}{dx} = x^2 - 2y\), and the given exact solution is \(y(x) = \frac{x^2}{2} - \frac{x}{2} + \frac{1}{4} + \frac{11}{4} \exp(-2x)\). Calculate \(\frac{dy}{dx}\) and confirm that it satisfies the differential equation:Calculate the derivative of \(y(x)\):\[\frac{d}{dx}\left(\frac{x^2}{2} - \frac{x}{2} + \frac{1}{4} + \frac{11}{4}\exp(-2x)\right) = x - \frac{1}{2} - \frac{11}{2} \exp(-2x)\cdot(-2)\]Simplifying, we get\[\frac{dy}{dx} = x - \frac{1}{2} + 11\exp(-2x)\]Now check the equation \(x^2 - 2y\):\[x^2 - 2 \left( \frac{x^2}{2} - \frac{x}{2} + \frac{1}{4} + \frac{11}{4} \exp(-2x) \right) = x^2 - x + \frac{1}{2} - 11 \exp(-2x)\]Thus, \(\frac{dy}{dx} - (x^2 - 2y) = 0\), confirming it satisfies the differential equation. Also, confirm \(y(0) = 3\) using the exact solution, substituting \(x = 0\):\[\frac{0^2}{2} - \frac{0}{2} + \frac{1}{4} + \frac{11}{4} \exp(0) = 0 + 0 + 0.25 + 2.75 = 3\]

Step 2: Calculate Euler's Method Approximation

The formula for Euler's Method is \(y_1 = y_0 + F(x_0, y_0) \Delta x\). First, calculate \(F(x_0, y_0)\) and \(\Delta x\):Given:- \(x_0 = 0\), \(y_0 = 3\)- \(F(x, y) = x^2 - 2y\)- \(x_1 = \frac{1}{4}\)- \(\Delta x = x_1 - x_0 = \frac{1}{4} - 0 = \frac{1}{4}\)Calculate \(F(x_0, y_0) = 0^2 - 2 \times 3 = -6\).Use Euler's Formula:\[y_1 = 3 + (-6) \times \frac{1}{4} = 3 - \frac{3}{2} = \frac{3}{2}\]

Step 3: Calculate Improved Euler's Method Approximation

First, compute \(F(x_1, y_1)\):\(x_1 = \frac{1}{4}\), \(y_1 = \frac{3}{2}\)\[F(x_1, y_1) = (\frac{1}{4})^2 - 2 \times \frac{3}{2} = \frac{1}{16} - 3 = -\frac{47}{16}\]Calculate the average slope \(m_1\):\[m_1 = \frac{F(x_0, y_0) + F(x_1, y_1)}{2} = \frac{-6 + (-\frac{47}{16})}{2} = -\frac{143}{32}\]Use Improved Euler's Formula:\[z_1 = y_0 + m_1 \Delta x = 3 - \frac{143}{32} \times \frac{1}{4} = 3 - \frac{143}{128} = \frac{241}{128}\]

Step 4: Compare Approximations with Exact Solution

Determine the exact value \(y(\frac{1}{4})\):\[y(\frac{1}{4}) = \frac{(\frac{1}{4})^2}{2} - \frac{\frac{1}{4}}{2} + \frac{1}{4} + \frac{11}{4}\exp(-2 \times \frac{1}{4})\]This simplifies to:\[y(\frac{1}{4}) = \frac{1}{32} - \frac{1}{8} + \frac{1}{4} + \frac{11}{4} \times \frac{1}{e^{1/2}} \approx 2.767\]Comparing \(y_1 = \frac{3}{2} = 1.5\) and \(z_1 = \frac{241}{128} \approx 1.882\), \(z_1\) is much closer to \(2.767\) than \(y_1\). Thus, the Improved Euler Method approximation \(z_1\) is more accurate.

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In simpler terms, they describe how a certain quantity changes over time. These equations are paramount in many fields, such as physics, engineering, and economics, because they often model the dynamics of systems. For instance, a differential equation might describe how the population of a city grows or how temperature changes within a material when heat is applied.

In the equation you'll encounter in this exercise, one variable, usually denoted as \( y \), depends on another variable, typically \( x \), and their relationship is described through derivatives—rates of change. The equation given here is \( \frac{dy}{dx} = x^2 - 2y \), meaning the rate at which \( y \) changes relative to \( x \) is determined by the formula on the right-hand side. Understanding how to work with such equations allows us to predict future scenarios or understand past behavior based on known data.

Initial Value Problem

An initial value problem (IVP) is a type of differential equation coupled with a specific condition at a given point. The purpose is to find a function that satisfies the differential equation and the initial condition. In this exercise, the equation is \( \frac{dy}{dx} = x^2 - 2y \) with the initial condition \( y(0) = 3 \). This tells us that when \( x = 0 \), the value of \( y \) must be 3. This point is essential for determining the unique solution to the differential equation.

Initial value problems are crucial in applications where you need to predict future outcomes from a known starting point, like modeling an object’s velocity falling under gravity or determining the current state in a circuit when the power is turned on. What makes them especially fascinating is that, depending on the given differential equation and starting conditions, they can yield a unique and precise solution that describes the behavior of the entire system.

Euler's Method

Euler’s Method is a straightforward numerical technique used to approximate solutions to initial value problems. It’s especially useful when a differential equation is difficult or impossible to solve analytically. The method works by using a derivative at a known point to estimate the value of a function at a nearby point.

Here’s a step-by-step view:

• Start at an initial point \( (x_0, y_0) \).
• Calculate the slope \( F(x_0, y_0) \) using the differential equation.
• Predict the new point \( y_1 = y_0 + F(x_0, y_0) \Delta x \).

The process involves successive small steps \( \Delta x \), where each new approximation relies on the previous point's slope. In our exercise, the method provides an approximation \( y_1 = 1.5 \), but as basic as it is, Euler's Method can lead to significant errors over larger intervals or in steeper functions without adjustments, which is why more refined methods such as the Improved Euler Method are often used.

Improved Euler Method

The Improved Euler Method, also known as Heun’s method, is a refinement of the simple Euler’s Method that provides better accuracy by using an average slope over a small interval for calculation. This improvement is achieved by correcting the predicted value at the end of the step.

The method involves these steps:

• Calculate an initial estimate with the basic Euler's method: \( y_1 \).
• Determine the slope at the new point \( F(x_1, y_1) \).
• Compute the average of the initial and new slopes: \( m_1 = \frac{F(x_0, y_0) + F(x_1, y_1)}{2} \).
• Use this average slope to get a more accurate estimation: \( z_1 = y_0 + m_1 \Delta x \).

In our exercise, this method generates \( z_1 \approx 1.882 \), a value significantly closer to the exact solution \( \approx 2.767 \). This demonstrates its superiority over the basic Euler method due to its consideration of slopes at two points, leading to a middle-ground prediction that often results in more accuracy, particularly over small intervals.