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? $$ y^{\prime}=2 t, y(0)=0 $$

Short answer

Euler's method gives an estimate of 0.8, while the exact solution at \( t=1 \) is 1, leading to an absolute error of 0.2.

Step-by-step solution

Understanding the Problem

We are given the differential equation \( y' = 2t \) with the initial condition \( y(0) = 0 \). Our task is to use Euler's method with \( n=5 \) steps over the interval \( t = [0,1] \) to estimate the solution, and then compare this estimate with the exact solution.

Preparation for Euler's Method

To apply Euler's method, we first calculate the step size \( h \). Given the interval \( [0,1] \) and \( n = 5 \) steps, the step size is \( h = \frac{1-0}{5} = 0.2 \). We will start at \( (t_0, y_0) = (0, 0) \).

Euler's Method Iteration 1

For the first step, compute \( y_1 = y_0 + h \cdot f(t_0, y_0) \), where \( f(t, y) = 2t \). Thus, \( y_1 = 0 + 0.2 \cdot 2\cdot0 = 0 \).

Euler's Method Iteration 2

Compute \( y_2 = y_1 + h \cdot f(t_1, y_1) \) with \( t_1 = 0.2 \). Therefore, \( y_2 = 0 + 0.2 \cdot 2\cdot0.2 = 0.08 \).

Euler's Method Iteration 3

Compute \( y_3 = y_2 + h \cdot f(t_2, y_2) \) where \( t_2 = 0.4 \). So, \( y_3 = 0.08 + 0.2 \cdot 2\cdot0.4 = 0.24 \).

Euler's Method Iteration 4

Compute \( y_4 = y_3 + h \cdot f(t_3, y_3) \) where \( t_3 = 0.6 \). Thus, \( y_4 = 0.24 + 0.2 \cdot 2\cdot0.6 = 0.48 \).

Euler's Method Iteration 5

Compute \( y_5 = y_4 + h \cdot f(t_4, y_4) \) where \( t_4 = 0.8 \). Consequently, \( y_5 = 0.48 + 0.2 \cdot 2\cdot0.8 = 0.8 \).

Finding the Exact Solution

Solve the differential equation \( y' = 2t \) exactly by integrating. The integral of \( 2t \) is \( t^2 + C \). Given \( y(0) = 0 \), we find \( C = 0 \). Thus, the exact solution is \( y = t^2 \).

Comparing with Exact Solution

Evaluate the exact solution at \( t = 1 \), which is \( y = 1^2 = 1 \). Compare this with the Euler's estimate \( y_5 = 0.8 \).

Accuracy Assessment

Euler's method with \( n = 5 \) estimates \( y(1) \) to be 0.8, while the exact value is 1. Thus, the absolute error is \( |0.8 - 1| = 0.2 \). Euler's method gives a rough estimate, which can be improved by increasing \( n \).

Key concepts

Differential Equations

Differential equations describe how one variable changes in relation to another variable. They are equations involving a function and its derivatives. These equations play a crucial role in modeling real-world phenomena, such as the growth of populations, the spread of diseases, or even how an object moves under the influence of forces.
For example, in our original exercise, we have the differential equation \( y' = 2t \), which implies that the rate of change of \( y \) with respect to \( t \) is proportional to \( t \) itself. Understanding this relationship is key to solving differential equations.
Solving a differential equation means finding the function \( y(t) \) that satisfies the equation. Depending on the complexity of the equation, finding an exact solution can range from being straightforward to extremely challenging. In our case, we solved it exactly to show \( y = t^2 \), which is derived by integrating the equation given.

Initial Value Problems

Initial value problems in the context of differential equations involve finding a particular solution given an initial condition. An initial condition is a specific point that the solution must pass through, often written as \( y(t_0) = y_0 \). This information is crucial because it eliminates the arbitrary constant that comes from solving differential equations.

In our example, we are given \( y(0) = 0 \) as the initial condition. This means that when \( t = 0 \), \( y \) must be zero. This condition guarantees a single, specific solution instead of a family of possible solutions. By applying the initial condition to the equation \( y = t^2 + C \), we obtain \( C = 0 \), yielding the exact solution \( y = t^2 \). Initial value problems are typically encountered in real-world applications, where initial conditions represent the state of a system at a starting time.

Numerical Methods

Numerical methods are approaches for finding approximate solutions to mathematical problems that cannot be solved analytically. These methods are particularly useful for complex differential equations where exact solutions are either difficult or impossible to find. Euler's method is one such numerical technique, commonly used to estimate solutions to initial value problems.

Euler's method involves estimating the solution by using linear approximations at a sequence of points. In the exercise, we divided the interval \( t = [0,1] \) into 5 steps and utilized the recursive formula \( y_{n+1} = y_n + h \, f(t_n, y_n) \) to calculate successive values of \( y \). Each step gives a closer approximation to the real solution.

• The step size \( h \) determines the accuracy of Euler's method; smaller step sizes generally yield better approximations.
• Using only 5 steps resulted in a relatively large error of 0.2 compared to the exact solution \( y = 1 \) at \( t=1 \).
• Increasing the number of steps would improve the accuracy.

Thus, Euler's method demonstrates how numerical methods can provide practical estimates where exact solutions are not feasible.