Question

Estimate the following solutions using Euler's method with \(n=5\) steps over the interval \(t=[0,1] .\) If you are able to solve the initialvalue problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler's method. How accurate is Euler's method? $$ y^{\prime}=y+t^{2}, y(0)=3 \text { . Exact solution is } y=5 e^{t}-2-t^{2}-2 t $$

Short answer

Euler's method approximated \( y(1) \) as 7.516, while the exact solution is about 8.59, with an error of approximately 1.07.

Step-by-step solution

Understand Euler's Method

Euler's method is a numerical procedure to solve ordinary differential equations (ODEs) with a given initial value. For the differential equation \( y' = f(t, y) \), and initial condition \( y(t_0) = y_0 \), Euler's method approximates the solution over an interval by advancing in steps, \( h \), using the formula: \[ y_{n+1} = y_n + h \, f(t_n, y_n) \] where \( t_n \) is the current time step and \( y_n \) is the current approximate solution.

Set Up Parameters for Euler's Method

We are given \( y' = y + t^2 \), \( y(0) = 3 \), with \( n = 5 \) steps over the interval \([0, 1]\). The step size \( h = \frac{1-0}{5} = 0.2 \). Initial values are \( t_0 = 0 \), \( y_0 = 3 \).

Calculate Iterations Using Euler's Method

Using the initial values and Euler's formula, we iterate as follows:* **Step 1:** - \( t_1 = 0 + 0.2 = 0.2 \) - \( y_1 = 3 + 0.2(3 + 0^2) = 3 + 0.6 = 3.6 \) * **Step 2:** - \( t_2 = 0.2 + 0.2 = 0.4 \) - \( y_2 = 3.6 + 0.2(3.6 + 0.2^2) = 3.6 + 0.752 = 4.352 \)* **Step 3:** - \( t_3 = 0.4 + 0.2 = 0.6 \) - \( y_3 = 4.352 + 0.2(4.352 + 0.4^2) = 4.352 + 0.8928 = 5.2448 \)* **Step 4:** - \( t_4 = 0.6 + 0.2 = 0.8 \) - \( y_4 = 5.2448 + 0.2(5.2448 + 0.6^2) = 5.2448 + 1.04896 = 6.29376 \)* **Step 5:** - \( t_5 = 0.8 + 0.2 = 1 \) - \( y_5 = 6.29376 + 0.2(6.29376 + 0.8^2) = 6.29376 + 1.221952 = 7.515712 \)

Calculate the Exact Solution at t=1

The given exact solution is \( y = 5e^t - 2 - t^2 - 2t \). Substitute \( t = 1 \) to find the exact value:\[ y(1) = 5e^1 - 2 - 1^2 - 2 \times 1 = 5e - 5 \]Using approximate value \( e \approx 2.718 \), the exact solution at \( t=1 \) is \( 5 \times 2.718 - 5 \approx 8.59 \).

Compare Approximate and Exact Solutions

From Step 3, the Euler's method approximation at \( t=1 \) is \( y_5 = 7.515712 \). The exact solution is approximately \( 8.59 \). Thus, the error is \( 8.59 - 7.515712 = 1.074288 \).

Determine Accuracy of Euler's Method

Euler's method with 5 steps approximated \( y(1) \) as 7.515712, deviating by about 1.074288 from the exact solution 8.59. The discrepancy highlights that while Euler's method is simple, it can accumulate significant error if step sizes are large for rapid changes in the function.

Key concepts

Ordinary Differential Equations (ODEs)

Ordinary Differential Equations (ODEs) are equations involving a function of one independent variable and its derivatives. These equations are fundamental in understanding dynamics across various fields like physics, biology, and engineering. The main goal when dealing with an ODE is to determine the function that satisfies the equation.

An ODE is typically expressed in the form: \( y' = f(t, y) \) where \( y \) is the dependent variable, \( t \) represents the independent variable (often time), and \( y' \) is the derivative of \( y \) with respect to \( t \).

ODEs can be classified based on their order, with the order representing the highest derivative present. ODEs are prevalent in modeling the behavior of changing systems, including oscillations, growth rates, and many more dynamic processes.

Numerical Solutions

Many ordinary differential equations cannot be solved using analytical methods. This is where numerical solutions come in handy. Instead of finding the exact solution, these methods provide approximate solutions.

**Why Use Numerical Solutions?**

• Some ODEs are too complex for an exact solution.
• They allow solving over any desired interval with flexibility.
• Numerical methods are highly adaptable and can handle various initial and boundary conditions.

Euler's Method is a straightforward numerical technique to approximate solutions of ODEs. It uses a step-by-step approach, calculating the slope at each step based on the current position and advancing the solution by a small amount, known as the step size \( h \).

This method is iterative, and its simplicity makes it a great introduction to understanding numerical approximations in solving differential equations.

Initial Value Problems

In many practical situations, we deal with initial value problems where the state of the system at the starting point is known. These problems specify an ordinary differential equation coupled with an initial condition.

For instance, given the ODE:
\( y' = y + t^2 \) with initial conditions \( y(0) = 3 \), we are tasked to find the function \( y(t) \) that satisfies these conditions.

Initial value problems are crucial because they form the basis for many real-world applications. Whether it's predicting population growth, the path of planets, or changes in financial markets, the conditions at the starting point impact the entire trajectory of the solution. Understanding initial value problems allow us to predict future states of dynamic systems accurately.

Accuracy of Numerical Methods

Accuracy in numerical methods refers to how close the approximate solution is to the exact solution of an ODE. Euler's Method, while simple, often gives a basic approximation with a certain degree of error, especially if the step size is not chosen well.

Several factors impact the accuracy:

• Step Size: A smaller step size \( h \) often yields better accuracy but requires more computations.
• Number of Steps: More steps might lead to cumulative errors if the method is not stable.
• Function Behavior: Functions with rapid changes can result in larger errors.

In our exercise, using 5 steps over the interval \([0, 1]\), Euler's method resulted in a noticeable error compared to the exact solution. Although Euler's method is easy to implement, for precise applications, it's often necessary to use more sophisticated methods that reduce error effectively, such as Runge-Kutta methods.