Question

Two steps of Euler's method 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. $$y^{\prime}(t)=-y, y(0)=-1 ; \Delta t=0.2$$

Short answer

Question: Compute the first two approximations for the given initial value problem using Euler's method with a time step of 0.2: \(y'(t) = -y\) and \(y(0) = -1\). Answer: The first two approximations using Euler's method for the given initial value problem are: $$u_1 = -0.8$$ $$u_2 = -0.64$$

Step-by-step solution

Understand Euler's method formula

To apply Euler's method, we need to use the following general formula: $$u_{n+1} = u_n + \Delta t \cdot f(t_n, u_n)$$ Here, \(f(t_n, u_n)\) represents the given differential equation, with an initial condition \(u_0 = y(0)\), and a time step \(\Delta t\). For the given problem, we have: $$f(t, y) = -y$$ and $$y(0) = -1$$

Calculate u1 (First Approximation)

Using the formula and known values, let's find the first approximation \(u_1\). $$u_1 = u_0 + \Delta t \cdot f(t_0, u_0)$$ We have: $$u_0 = y(0) = -1$$ $$\Delta t = 0.2$$ $$f(t_0, u_0) = -u_0 = 1$$ Now, plug in the values to get: $$u_1 = -1 + 0.2 \cdot 1 = -1 + 0.2 = -0.8$$

Calculate u2 (Second Approximation)

Next, we have to find the second approximation \(u_2\). Use the formula again with the new values. $$u_2 = u_1 + \Delta t \cdot f(t_1, u_1)$$ Note that: $$t_1 = 0.2$$ $$u_1 = -0.8$$ $$f(t_1, u_1) = -u_1 = 0.8$$ Plugging in the values, we get: $$u_2 = -0.8 + 0.2 \cdot 0.8 = -0.8 + 0.16 = -0.64$$ Thus, the first two approximations using Euler's method for the given initial value problem are: $$u_1 = -0.8$$ $$u_2 = -0.64$$

Key concepts

Initial Value Problems

In the realm of differential equations, an Initial Value Problem (IVP) is essentially a puzzle that gives you a starting point and a rule for finding the path ahead. It usually consists of an ordinary differential equation (ODE) and an accompanying initial condition. This initial condition pinpoints the value of the unknown function at a specific point. In our problem, the differential equation is given by \( y'(t) = -y \), and the initial condition is \( y(0) = -1 \). This condition helps anchor the solution curve. Without it, we could have infinitely many potential solutions since differential equations sometimes have families of solutions. The role of Euler's method in an IVP is to approximate these solutions numerically instead of solving them analytically, especially when an algebraic solution is hard or impossible to find. Thus, the initial values provide the essential groundwork from which numerical methods can begin their approximation efforts.

Numerical Approximation

Numerical Approximation involves using mathematical techniques to find approximate solutions to mathematical problems that may not be easy to solve exactly. Euler's method is a basic—yet powerful—technique for doing this with differential equations, particularly when dealing with initial value problems. In our exercise, Euler's method gives us approximate solutions \( u_1 \) and \( u_2 \) as steps to solving the differential equation using a specified time step \( \Delta t = 0.2 \). Instead of calculating the exact continuous trajectory of the solution, Euler's method incrementally progresses in small steps, predicting the new values based on current information. It uses a simple formula:

• \( u_{n+1} = u_n + \Delta t \cdot f(t_n, u_n) \)

With each step, we obtain a new approximation. This iterative nature can introduce some errors, but it also allows us to gain insights into the nature of solutions without full-on solving the differential equation analytically. The smaller the time step, generally, the closer the approximation will be to the actual solution.

Differential Equations

Differential Equations are a fundamental tool in modeling real-world phenomena, where rates of change play a significant role. A differential equation relates a function to its derivatives, essentially describing how a particular quantity changes over time or space. In our example, \( y'(t) = -y \) speaks to the rate of change of \( y \); it tells us that the rate is proportional to the negative of its current value. This type of equation frequently appears in scenarios where processes decay exponentially or substances cool down. Solving differential equations analytically can sometimes be straightforward, as in simple linear cases, but they often require numerical methods like Euler's method for complex or nonlinear situations. Understanding how to set up a differential equation helps decode and interpret the puzzles they present, providing solutions that describe dynamic systems and their behavior over time.