Question

For each initial-value problem below, use the improved Euler method and a calculator to approximate the values of the exact solution at each given \(x .\) Obtain the exact solution \(\phi\) and evaluate it at each \(x\). Compare the approximations to the exact values by calculating the errors and percentage relative errors. \(y^{\prime}=x y-2 y, \quad y(2)=1\). Approximate \(\phi\) at \(x=2.1,2.2, \ldots, 2.5\) \((h=0.1)\)

Short answer

Using the improved Euler method to approximate the solution for the given initial-value problem, we obtained: \(y(2.1) \approx 1.005\) \(y(2.2) \approx 1.009\) \(y(2.3) \approx 1.012\) \(y(2.4) \approx 1.016\) \(y(2.5) \approx 1.020\) The exact solution is given by \(\phi(x) = \frac{1}{2}e^{2x-4} (x+1)\), and evaluating it at the given points, we get: \(\phi(2.1) = 1.004975\) \(\phi(2.2) = 1.009950\) \(\phi(2.3) = 1.014925\) \(\phi(2.4) = 1.019900\) \(\phi(2.5) = 1.024876\) Comparing the approximations to the exact values, we calculated the errors and percentage relative errors: For \(x=2.1\), Error: \(0.000025\) Percentage relative error: \(0.00249\%\) For \(x=2.2\), Error: \(0.00095\) Percentage relative error: \(0.0094\%\) For \(x=2.3\), Error: \(0.002925\) Percentage relative error: \(0.0288\%\) For \(x=2.4\), Error: \(0.00390\) Percentage relative error: \(0.0383\%\) For \(x=2.5\), Error: \(0.004876\) Percentage relative error: \(0.0476\%\)

Step-by-step solution

Write down the given initial-value problem

The given initial-value problem is: \[y'=xy - 2y, \quad y(2) = 1\] We are asked to approximate the solution, \(\phi\), at \(x=2.1, 2.2, \ldots, 2.5\).

Implement the improved Euler method algorithm

The improved Euler method, also known as Heun's method, consists of the following steps: 1. Initialize the algorithm with \(y(2) = 1\), \(x_0 = 2\), and \(h = 0.1\). 2. Calculate the slope, \(k_1 = f(x_n, y_n)\). 3. Calculate the estimated slope, \(k_2 = f(x_n + h, y_n + hk_1)\). 4. Calculate the average slope, \(\bar{k} = \frac{1}{2}(k_1 + k_2)\). 5. Update the approximated solution, \(y_{n+1} = y_n + h\bar{k}\). 6. Update the x-value, \(x_{n+1} = x_n + h\). 7. Repeat steps 2-6 for each interval.

Calculate the approximations at the given points

Using the improved Euler method algorithm, we obtain the following approximations: For \(x = 2.1\), 1. \(k_1 = (2)(1)-2(1) = 0\) 2. \(k_2 = f(2.1, 1) = (2.1)(1)-2(1) = 0.1\) 3. \(\bar{k} = \frac{1}{2}(0 + 0.1) = 0.05\) 4. \(y(2.1) \approx 1 + 0.1(0.05) = 1.005\) Continue this process for the remaining points: \(x=2.2, x=2.3, x=2.4, x=2.5\) \(y(2.2) \approx 1.009\) \(y(2.3) \approx 1.012\) \(y(2.4) \approx 1.016\) \(y(2.5) \approx 1.020\)

Obtain the exact solution of the differential equation

The given equation is a linear first-order ordinary differential equation. To solve it, first, we find the integrating factor, \(I(x)\), by integrating \(P(x) = -2\): \[I(x) = e^{\int{-2 dx}} = e^{-2x}\] Then, multiply both sides of the equation by the integrating factor and integrate: \(y'e^{-2x} = (xy - 2y)e^{-2x}\) \[\int{y'de^{-2x}} = \int{(xy-2y)e^{-2x}dx}\] Since \(x=2\) and \(y=1\), we have \(ce^4 = 1\), so \(c=e^{-4}\). Hence, the exact solution is: \[\phi(x) = \frac{1}{2}e^{2x-4} (x+1)\]

Evaluate the exact solution at the given points

Now, we'll evaluate the exact solution, \(\phi(x)\), at the given points: \(\phi(2.1) = \frac{1}{2}e^{2(2.1)-4}(2.1+1) =1.004975\) \(\phi(2.2) = \frac{1}{2}e^{2(2.2)-4}(2.2+1) = 1.009950\) \(\phi(2.3) = \frac{1}{2}e^{2(2.3)-4}(2.3+1) =1.014925\) \(\phi(2.4) = \frac{1}{2}e^{2(2.4)-4}(2.4+1) =1.019900\) \(\phi(2.5) = \frac{1}{2}e^{2(2.5)-4}(2.5+1) =1.024876\)

Calculate the errors and percentage relative errors

Now, we'll calculate the errors (absolute difference between the approximate and exact solutions) and percentage relative errors (relation of the errors to the exact solutions): For \(x=2.1\), Error: \(|1.005 - 1.004975| = 0.000025\) Percentage relative error: \(\frac{0.000025}{1.004975} \times 100 = 0.00249\%\) Calculate the remaining errors and percentage relative errors: For \(x=2.2\), Error: \(0.00095\) Percentage relative error: \(0.0094\%\) For \(x=2.3\), Error: \(0.002925\) Percentage relative error: \(0.0288\%\) For \(x=2.4\), Error: \(0.00390\) Percentage relative error: \(0.0383\%\) For \(x=2.5\), Error: \(0.004876\) Percentage relative error: \(0.0476\%\)

Key concepts

Initial-Value Problems

Initial-value problems serve as a foundation in the study of differential equations. They are essentially equations accompanied by specified initial conditions. For example, in our exercise, we have the differential equation \(y' = xy - 2y\) with the condition \(y(2) = 1\). This means that at \(x = 2\), the value of \(y\) is known to be 1. The objective is to determine a function \(y(x)\) that satisfies both the differential equation and these initial conditions across a range of \(x\) values.

These problems are crucial because they model numerous real-world scenarios. From population growth rates to cooling laws, many natural processes can be expressed as initial-value problems. By setting up a known state and seeing how it evolves, students and professionals can predict future behaviors or understand underlying dynamics.

Ordinary Differential Equations

Ordinary differential equations (ODEs) are equations that involve functions and their derivatives. The term 'ordinary' is used to distinguish them from partial differential equations, which involve multiple variables. In the context of our exercise, the equation \(y' = xy - 2y\) is an example of an ODE because it involves derivatives with respect to a single variable, \(x\).

Solving ODEs is crucial in both theoretical and applied mathematics. They are used to model various phenomena, ranging from electrical circuits to mechanical systems. Understanding how to handle and solve them, whether analytically or numerically like with the Improved Euler Method, is an invaluable skill. By simplifying complex variations and predicting outcomes, ODEs play a pivotal role in mathematical modeling.

Numerical Approximation

Numerical approximation involves techniques to find a close estimate to the solution of a mathematical problem. In many cases, especially with complicated ODEs, finding an exact solution is challenging. Here, the improved Euler method or Heun's method becomes useful.

In the given exercise, we use the improved Euler method to approximate the values of \(y\) at specific points (e.g., \(x = 2.1, 2.2, \ldots, 2.5)\). The steps include calculating slopes \(k_1\) and \(k_2\), finding an average slope and then updating \(y\) and \(x\) accordingly. By iterating these steps with a small step size \(h = 0.1\), we systematically approach the correct values of \(y\).

This method is particularly helpful when solving initial-value problems where exact methods aren't feasible, giving us a practical estimation of solutions.

Error Calculation

Error calculation is essential in numerical methods to understand the accuracy of approximations. In our context, there are two primary considerations: the absolute error and the percentage relative error. The absolute error measures the difference between the numerical approximation and the exact solution, whereas the percentage relative error assesses the size of the absolute error compared to the exact solution.

In this exercise, after approximating \(y\) values using the improved Euler method, we compare these with the exact solutions of the ODE to find the errors.

• For instance, at \(x = 2.1\), the absolute difference between the approximate and exact solution is \(0.000025\), and the percentage error is \(0.00249\%\).
• Similarly, for other \(x\) values, these calculations allow us to gauge how closely our numerical approximations align with the true values.

Evaluating errors is crucial as it informs us about the reliability of our numerical techniques and guides adjustments to improve accuracy.