Question

For the following initial value problems, compute the first two approximations \(u_{1}\) and \(u_{2}\) given by Euler's method using the given time step. $$y^{\prime}(t)=t+y, y(0)=4 ; \Delta t=0.5$$

Short answer

Question: Compute the first two approximations of the first-order initial value problem using Euler's method with a time step of 0.5. Given: $$y^{\prime}(t)=t+y, y(0)=4$$ Answer: Using Euler's method with a time step of 0.5, the first two approximations for the given initial value problem are \(u_1=6\) and \(u_2=9.25\).

Step-by-step solution

Initial values

First, identify the given information from the problem. The initial value is \(y(0)=4\), the differential equation is \(y^{\prime}(t)=t+y\), and the time step is \(\Delta t=0.5\).

Euler's method formula

The general formula for Euler's method is given by: $$y_{n+1}=y_n+\Delta t f(t_n,y_n)$$ where \(y_n\) is the approximate value of the function at time \(t_n = n\Delta t\), \(\Delta t\) is the step size, and \(f(t_n,y_n)\) represents the derivative with respect to \(t\) based on the given ODE. In our case, \(f(t,y) = t+y\).

Calculate \(u_1\)

For the first approximation, \(u_1\), we have: \begin{align*} y_1 &= y_0 + \Delta t f(t_0,y_0) \\ &= 4 + 0.5 (0 + 4) \\ &= 4 + 2 \\ &= 6 \end{align*} So \(u_1 = 6\).

Calculate \(u_2\)

For the second approximation, \(u_2\), we have: \begin{align*} y_2 &= y_1 + \Delta t f(t_1,y_1) \\ &= 6 + 0.5 (0.5 + 6) \\ &= 6 + 0.5 (6.5) \\ &= 6 + 3.25 \\ &= 9.25 \end{align*} So \(u_2 = 9.25\).

Conclusion

Using Euler's method with a time step of \(\Delta t=0.5\), our first two approximations for the given initial value problem are \(u_1=6\) and \(u_2=9.25\).

Key concepts

Differential Equations

Differential equations are mathematical expressions that involve derivatives, a type of function that describes changes within a system. These equations are crucial because they allow us to model and predict real-world phenomena, from physics to engineering and beyond. At the heart of these equations, you will find a function and its derivative. For example, in the exercise we considered, the differential equation is given by \(y'(t) = t + y\). This expresses how the rate of change of \(y\) depends on both the variable \(t\) and the function \(y\) itself.

By solving differential equations, you can understand how systems evolve over time. There are many types of differential equations, such as ordinary, partial, linear, and nonlinear. In the context here, we focus on ordinary differential equations (ODEs) because they involve functions of a single variable, like time.

Understanding how to manage and solve these equations is critical, especially when analytical (exact) solutions are hard or impossible to find. This is where numerical methods, like Euler's method, step in to offer approximations.

Numerical Methods

Numerical methods are techniques used to approximate solutions to problems that may otherwise be unsolvable by traditional analytical methods. These methods are especially useful for complex differential equations where finding an exact solution could be difficult.

Euler's method, which we applied in the exercise, is one of the simplest and most straightforward numerical methods. It provides a method to approximate solutions of differential equations by using stepwise calculations. Here's a quick look at how it works:

• You start with an initial value.
• You calculate the slope (using the derivative given by the differential equation) at that initial point.
• You then use this slope to estimate the next value in the sequence.

In our exercise, we computed the next values using \(y_{n+1} = y_n + \Delta t \cdot f(t_n, y_n)\), where \(f(t,y) = t+y\). This technique gives us a glimpse into how the solution evolves with each step, opening the door to understanding more complex behaviors of the system being modeled.

Initial Value Problems

Initial value problems (IVPs) are a specific type of differential equation problem where the solution is determined based on a given starting point, known as the initial condition. This initial condition is essential because it sets the stage for solving the problem. Simply put, it dictates the path the solution will take. In the given exercise, the IVP is \(y(0) = 4\).

Solving an IVP involves using the differential equation along with the initial condition to find an approximate or exact solution. Euler's method, which we utilize here, excels at providing approximations for the solutions of these problems when analytical solutions are not readily available or are too complex to derive.

With Euler's method, you essentially "build" the solution step-by-step. Starting from the initial condition, each subsequent step relies on calculations made in the previous one. This sequential dependency means that the initial value is crucial to the accuracy and relevance of the solution obtained. Thus, handling initial value problems adeptly through numerical methods like Euler's methods is invaluable, especially in scientific and engineering applications where models need to be computed and predicted efficiently.