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? $$ y^{\prime}=-5 t, y(0)=-2 . \text { Exact solution is } y=-\frac{5}{2} t^{2}-2 $$

Short answer

Euler's method approximated the solution with increasing error; the maximum error was 0.5 at \( t = 1.0 \).

Step-by-step solution

Identifying Initial Conditions and Setup

Euler's method is an iterative procedure for solving ordinary differential equations (ODEs). The given differential equation is \( y' = -5t \) and the initial condition is \( y(0) = -2 \). We will use Euler's method with \( n = 5 \) over the interval \([0, 1]\). This means \( h = \frac{1}{5} = 0.2 \). We need to calculate \( y(t) \) at each step using the formula: \( y_{n+1} = y_n + h \cdot f(t_n, y_n) \).

Calculate Step Size

The interval \([0, 1]\) is divided into \(n = 5\) equal parts, so the step size is calculated as \( h = \frac{1 - 0}{5} = 0.2 \).

Compute the Iterative Steps of Euler's Method

Let's calculate step-by-step:1. **At \( t = 0 \):** - Initial: \( y_0 = -2 \) - Slope \( f(t_0, y_0) = -5 \cdot 0 = 0 \) - \( y_1 = -2 + 0.2 \cdot 0 = -2 \)2. **At \( t = 0.2 \):** - \( y_1 = -2 \) - Slope \( f(t_1, y_1) = -5 \cdot 0.2 = -1 \) - \( y_2 = -2 + 0.2 \cdot (-1) = -2.2 \)3. **At \( t = 0.4 \):** - \( y_2 = -2.2 \) - Slope \( f(t_2, y_2) = -5 \cdot 0.4 = -2 \) - \( y_3 = -2.2 + 0.2 \cdot (-2) = -2.6 \)4. **At \( t = 0.6 \):** - \( y_3 = -2.6 \) - Slope \( f(t_3, y_3) = -5 \cdot 0.6 = -3 \) - \( y_4 = -2.6 + 0.2 \cdot (-3) = -3.2 \)5. **At \( t = 0.8 \):** - \( y_4 = -3.2 \) - Slope \( f(t_4, y_4) = -5 \cdot 0.8 = -4 \) - \( y_5 = -3.2 + 0.2 \cdot (-4) = -4 \)

Compare Euler's Method with Exact Solution

Calculate the exact solution for the same points using the provided exact formula \( y = -\frac{5}{2}t^2 - 2 \).1. **At \( t = 0.2 \):** - Exact: \( y = -\frac{5}{2}(0.2)^2 - 2 = -2.1 \) - Error: \( | -2.1 - (-2.2) | = 0.1 \)2. **At \( t = 0.4 \):** - Exact: \( y = -\frac{5}{2}(0.4)^2 - 2 = -2.4 \) - Error: \( | -2.4 - (-2.6) | = 0.2 \)3. **At \( t = 0.6 \):** - Exact: \( y = -\frac{5}{2}(0.6)^2 - 2 = -2.9 \) - Error: \( | -2.9 - (-3.2) | = 0.3 \)4. **At \( t = 0.8 \):** - Exact: \( y = -\frac{5}{2}(0.8)^2 - 2 = -3.6 \) - Error: \( | -3.6 - (-4) | = 0.4 \)5. **At \( t = 1.0 \):** - Exact: \( y = -\frac{5}{2}(1.0)^2 - 2 = -4.5 \) - Euler's final: \( y_5 = -4 \) - Error: \( | -4.5 - (-4) | = 0.5 \)

Accuracy Evaluation

Euler's method provides an approximation, which becomes less accurate as the interval increases. The maximum observed error between Euler's method and the exact solution in this case is 0.5 at \( t = 1.0 \). The error tends to increase with each step because small errors add up cumulatively.

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations that involve a function and its derivatives. They are essential for modeling the dynamics of systems in physics, engineering, and other sciences. An ODE describes how a quantity changes with respect to another, usually time. For example, in our original problem, we have the ODE \( y' = -5t \), which tells us how the rate of change of \( y \) is related to the variable \( t \).

These equations often come with initial conditions, which specify the value of the function at a certain point, aiding in finding a unique solution. In this problem, the initial condition is \( y(0) = -2 \).

Solving ODEs can often be complex, especially if they cannot be tackled by simple analytical methods. This is where numerical methods like Euler's Method come in handy. They provide an approach to approximate solutions for ODEs that are otherwise challenging to solve exactly.

Numerical methods

Numerical methods involve algorithms that can give approximate solutions to mathematical problems. These are particularly useful for solving equations that do not have straightforward analytical solutions, like many ODEs. Euler's Method is a basic numerical method for solving initial value problems for ODEs.

In Euler's Method, the idea is to use a series of small steps to reach the solution over an interval. This is done by approximating the slope of the solution curve at discrete points. For the problem \( y' = -5t \), Euler's Method uses the formula \( y_{n+1} = y_n + h \cdot f(t_n, y_n) \) to estimate the values at each step.

Despite its simplicity, Euler's Method can produce significant errors if not used with a sufficiently small step size, which is why other more advanced numerical methods have been developed to improve accuracy.

Initial value problems

Initial value problems (IVPs) are a specific type of problem where the value of the solution is known at a starting point. In these problems, one must solve the differential equation given this initial condition. In our exercise, the IVP is defined by the equation \( y' = -5t \) with the initial condition \( y(0) = -2 \).

The goal of solving an IVP is to find the function \( y(t) \) that not only satisfies the equation but also adheres to the initial condition. Numerical methods like Euler's Method are used to approximate \( y(t) \) over the desired interval. By iterating, starting at the initial condition, you can predict the behavior of the system described by the differential equation.

Such problems are crucial in evaluating real-world situations where the beginning conditions can significantly impact the outcomes, such as in engineering and physics applications.

Error analysis

Error analysis in numerical methods helps to understand the accuracy of the approximate solution. As numerical methods often give close estimates rather than exact answers, evaluating the closeness or error is vital. It can be computed as the difference between the numerical solution and the exact solution.

In our exercise, error is calculated at each step by comparing Euler's approximation with the exact solution \( y = -\frac{5}{2}t^2 - 2 \). For instance, when \( t = 0.2 \), the error is \( | -2.1 - (-2.2) | = 0.1 \).

The cumulative nature of numerical errors is significant. Small errors at each step can add up, leading to larger discrepancies, particularly over larger intervals. This example demonstrated that the error grows as \( t \) approaches 1.0, where it reaches a maximum of 0.5. Understanding the limitations and accuracy of Euler's Method is crucial for effectively applying it to solve ODEs.