Problem 59 - 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. $$ \begin{array}{l} d y / d x=x+y, y(1)=2, x_{1}=1.2 ; \text { Exact solution: } y(x)= \\ 4 \exp (x-1)-x-1 \end{array} $$

Step-by-Step Solution

Answer

The Improved Euler method approximation \( z_1 = 2.68 \) is more accurate than the Euler method approximation \( y_1 = 2.6 \).

Step 1: Verify the Exact Solution is Correct

The given differential equation is \( \frac{dy}{dx} = x + y \). The provided exact solution is \( y(x) = 4e^{x-1} - x - 1 \). First, compute the derivative of the solution: \( \frac{dy}{dx} = 4e^{x-1} - 1 \). Substitute both \( y(x) \) and \( \frac{dy}{dx} \) into the differential equation: \( 4e^{x-1} - 1 = x + (4e^{x-1} - x - 1) \). Simplifying the right side gives \( 4e^{x-1} - 1 \), thus confirming the solution satisfies the differential equation. Also, check the initial condition by substituting \( x = 1 \): \( y(1) = 4e^{1-1} - 1 - 1 = 2 \), which is correct.

Step 2: Compute Euler's Method Approximation

Using Euler's method, start with \( y_0 = 2 \) at \( x_0 = 1 \) and \( x_1 = 1.2 \), giving \( \Delta x = 0.2 \). Substitute into the function \( F(x_0, y_0) = x_0 + y_0 = 1 + 2 = 3 \). Compute \( y_1 = y_0 + F(x_0, y_0) \Delta x = 2 + 3 \times 0.2 = 2.6 \). So, \( y_1 = 2.6 \).

Step 3: Compute Improved Euler Method Approximation

In Improved Euler's Method, first calculate \( F(x_1, y_1) = x_1 + y_1 = 1.2 + 2.6 = 3.8 \). Then the average slope \( m_1 = \frac{F(x_0, y_0) + F(x_1, y_1)}{2} = \frac{3 + 3.8}{2} = 3.4 \). Now compute \( z_1 = y_0 + m_1 \Delta x = 2 + 3.4 \times 0.2 = 2.68 \). Hence, \( z_1 = 2.68 \).

Step 4: Compare with Exact Solution at x=1.2

Substituting \( x_1 = 1.2 \) into the exact solution yields \( y(1.2) = 4e^{1.2-1} - 1.2 - 1 \). Computing \( 4e^{0.2} - 2.2 \), first calculate \( e^{0.2} \approx 1.2214 \). Thus, \( y(1.2) = 4 \times 1.2214 - 2.2 = 4.8856 - 2.2 = 2.6856 \). The Improved Euler Method approximation \( z_1 = 2.68 \) is closer than the Euler Method approximation \( y_1 = 2.6 \) to the exact value \( 2.6856 \).

Key Concepts

Initial Value Problem

An Initial Value Problem (IVP) is a type of differential equation that comes with a specific condition. It provides a way to find a particular solution among all possible solutions to a differential equation by specifying a value at a starting point. For instance, in this exercise, we have the equation \( \frac{dy}{dx} = x + y \). The initial value is given as \( y(1) = 2 \). This initial condition helps to determine a unique solution that satisfies both the differential equation and the initial value condition. This particular problem gives us an exact solution \( y(x) = 4 \exp(x-1) - x - 1 \), which conforms perfectly with the initial condition, confirming the problem is solvable on this basis.
This concept ensures we're not just finding any function that satisfies the equation, but the specific one that begins at the given point \((x_0, y_0) = (1, 2)\).

Euler's Method

Euler's Method is a straightforward numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It's a simple way to approximate solutions by breaking the problem into small steps. Here's how it works in this exercise:

• Start with known values \(y_0 = 2\) at \(x_0 = 1\).
• Use \(F(x_0, y_0) = x_0 + y_0 = 3\) to calculate the next value.
• The change in \(x\) is \(\Delta x = 0.2\).
• Compute \(y_1 = y_0 + F(x_0, y_0) \Delta x = 2 + 3 \times 0.2 = 2.6\).

This method uses the tangent line at the starting point to estimate the function's value at a new point. Although it's simple, Euler's Method can be inaccurate over larger steps or complex equations. In this case, it gives us an approximation \(y_1 = 2.6\).

Improved Euler Method

The Improved Euler Method, sometimes known as the Heun's Method, enhances the basic Euler's Method by refining the estimate of the slope. This results in better accuracy with the same step size. Here's how it improves upon the Euler Method:

• Calculate slope at the beginning: \(F(x_0, y_0) = 3\).
• Estimate \(y_1\) using the simple Euler's Method to find the slope at the end: \(F(x_1, y_1) = 3.8\).
• Compute the average slope: \(m_1 = \frac{3 + 3.8}{2} = 3.4\).
• Use the average slope for a more precise approximation: \(z_1 = y_0 + m_1 \Delta x = 2 + 3.4 \times 0.2 = 2.68\).

By averaging the initial and the estimated final slopes, the Improved Euler Method typically results in a value closer to the exact answer than the simple Euler's Method.

Exact Solution Verification

Exact Solution Verification checks that the provided solution meets the problem's conditions. It involves substituting the solution back into the differential equation and initial condition to confirm correctness.
For this exercise:

• The given solution is \(y(x) = 4 \exp(x-1) - x - 1\).
• Its derivative is \(\frac{dy}{dx} = 4 \exp(x-1) - 1\).
• Substitute \(y(x)\) and \(\frac{dy}{dx}\) into the equation to verify both sides match.
• Ensure it satisfies the initial condition \(y(1) = 2\).

Verification confirms our solution is correct and adheres to the problem's constraints. The calculated exact value at \(x_1 = 1.2\) is \(y(1.2) = 2.6856\), which matches well with the improved prediction \(z_1 = 2.68\), showing how close the numerical method can come to the true solution.