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. $$y^{\prime}(t)=2-y, y(0)=1 ; \Delta t=0.1$$

Short answer

Question: Find the first two approximations, \(u_{1}\) and \(u_{2}\), using Euler's method for the initial value problem \(y'(t)=2-y\) with initial condition \(y(0) = 1\), and time step \(\Delta t = 0.1\). Answer: The first two approximations of the given initial value problem using Euler's method are \(u_{1} = 1.1\) and \(u_{2} = 1.19\).

Step-by-step solution

Find the function f(t,u)

For the given ODE, \(y'(t) = 2 - y\), the function f(t,u) is: $$f(t,u) = 2 - u$$

Calculate the first approximation u_{1}

Use the Euler's method formula: \begin{align*} u_{1} &= u_{0} + \Delta t \cdot f(t_0, u_0) \\ u_{1} &= 1 + 0.1(2 - 1) \\ u_{1} &= 1 + 0.1 \\ u_{1} &= 1.1 \end{align*} The first approximation, \(u_{1}\), is 1.1.

Calculate the second approximation u_{2}

Use the Euler's method formula again: \begin{align*} u_{2} &= u_{1} + \Delta t \cdot f(t_1, u_1) \\ u_{2} &= 1.1 + 0.1(2 - 1.1) \\ u_{2} &= 1.1 + 0.1(0.9) \\ u_{2} &= 1.1 + 0.09 \\ u_{2} &= 1.19 \end{align*} The second approximation, \(u_{2}\), is 1.19. Thus, the first two approximations of the given initial value problem using Euler's method are \(u_{1} = 1.1\) and \(u_{2} = 1.19\).

Key concepts

Initial Value Problem

An Initial Value Problem (IVP) is a type of differential equation that comes with an initial condition. The goal is to find a solution to the equation that satisfies this condition at a starting point. In our example, the problem includes a differential equation, \(y'(t) = 2 - y\), with an initial value \(y(0) = 1\). This means that at time \(t = 0\), the value of the function \(y\) is 1.

Problems like these involve finding a function \(y(t)\) that satisfies both the differential equation and the initial condition. Solving IVPs can be challenging because the function’s behavior is determined by how its rate of change and its initial condition interact as time progresses. Euler’s method, as illustrated, is a useful numerical tool to approximate the solution over short time steps.

Numerical Approximation

Numerical approximation is a technique used to find approximate solutions to complex mathematical problems that cannot be solved exactly. It becomes especially useful when dealing with differential equations. Euler's method is one such technique that provides a step-by-step approach to approximate the solution.

For the given initial value problem, numerically approximating \(y(t)\) helps us understand how the function behaves over time, without needing a precise mathematical solution. By doing short steps from the known starting point, Euler's method makes it easier to get a clear picture of the function's path. This is especially valuable in contexts where obtaining an analytical solution is difficult or impossible.

Ordinary Differential Equation

An Ordinary Differential Equation (ODE) is a mathematical equation involving a function and its derivatives. It describes the rate at which something changes over time or another independent variable. In our example, the ODE \(y'(t) = 2 - y\) shows how the rate of change of \(y\) depends on \(y\) itself.

ODEs are common in modeling real-world phenomena, such as population growth, radioactive decay, and engineering systems. Understanding how to solve or approximate solutions to ODEs is essential for these applications. They provide a framework for predicting future behavior based on current states and influences from other variables. Euler’s method simplifies this understanding by transforming the continuous change into small, manageable steps.

Time Step

A Time Step, denoted as \(\Delta t\), is the increment of time used to move from one approximation to the next in numerical methods like Euler's method. In our problem, the time step is 0.1. This means that each approximation is made by calculating the function's behavior over this small interval.

The size of the time step impacts the accuracy of the numerical approximation. Smaller time steps generally lead to more accurate results because they better capture the intricacies of the function's change. However, they also require more calculations. Euler's method uses the time step to iterate forward through the function, ensuring step-by-step transitions between approximations are both efficient and informative.