Question

Use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=-y, y(0)=2,[0,1] $$

Short answer

The approximate solution for \( y(1) \) using Euler's Method is \( 0.65536 \).

Step-by-step solution

Understanding the Problem

Our differential equation is given by \( y' = -y \) with an initial condition \( y(0) = 2 \). We need to approximate the solution using Euler's Method over the interval \([0, 1]\) with a step size \( h = 0.2 \).

Setting Up Euler's Method

Euler's Method is an iterative procedure for approximating solutions to differential equations. The formula to apply is \( y_{n+1} = y_n + h f(x_n, y_n) \), where \( f(x, y) = y' \). In our case, \( f(x, y) = -y \).

Calculating the First Step

We start with \( y_0 = 2 \) and \( x_0 = 0 \). First, we calculate the derivative at this point: \( f(x_0, y_0) = -y_0 = -2 \). Next, we update \( y \): \( y_1 = y_0 + h f(x_0, y_0) = 2 + 0.2(-2) = 2 - 0.4 = 1.6 \).

Calculating the Second Step

We now use \( y_1 = 1.6 \) and \( x_1 = 0.2 \). Calculate the new derivative: \( f(x_1, y_1) = -y_1 = -1.6 \). Update \( y \): \( y_2 = y_1 + h f(x_1, y_1) = 1.6 + 0.2(-1.6) = 1.6 - 0.32 = 1.28 \).

Calculating the Third Step

Now with \( y_2 = 1.28 \) and \( x_2 = 0.4 \). Calculate: \( f(x_2, y_2) = -1.28 \). Update \( y \): \( y_3 = y_2 + h f(x_2, y_2) = 1.28 + 0.2(-1.28) = 1.28 - 0.256 = 1.024 \).

Calculating the Fourth Step

Use \( y_3 = 1.024 \) and \( x_3 = 0.6 \). Find: \( f(x_3, y_3) = -1.024 \). Update: \( y_4 = y_3 + h f(x_3, y_3) = 1.024 + 0.2(-1.024) = 1.024 - 0.2048 = 0.8192 \).

Calculating the Fifth Step

Now use \( y_4 = 0.8192 \) and \( x_4 = 0.8 \). Calculate \( f(x_4, y_4) = -0.8192 \). Update: \( y_5 = y_4 + h f(x_4, y_4) = 0.8192 + 0.2(-0.8192) = 0.8192 - 0.16384 = 0.65536 \).

Final Approximation

With the final value \( y_5 = 0.65536 \), Euler's Method gives us the approximation for \( y(1) \) as approximately \( 0.65536 \).

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They are essential in modeling a variety of natural phenomena, such as population growth, heat distribution, and the motion of waves. In the given exercise, the differential equation is represented as \( y' = -y \). This relationship implies that the rate of change of the function \( y \) is proportional to the negative of the function itself. This specific type is known as a first-order linear differential equation.

Such equations describe situations where the quantity being modeled decreases exponentially over time or distance. Understanding differential equations is crucial for solving real-world problems where change occurs continuously. They provide the foundation for modeling motion, growth, and decay processes and are utilized extensively across physics, engineering, and other scientific disciplines.

Numerical Approximation

Numerical approximation is a technique used to find an approximate solution to mathematical problems that cannot easily be solved analytically. Euler's Method is a specific numerical approach for approximating solutions to differential equations when an exact solution is either difficult, or impossible, to obtain.

Euler's Method provides a step-by-step way to estimate the value of the unknown function at certain points. It calculates the solution at discrete points by incrementally proceeding from an initial condition along small steps, the results of which assemble an approximate solution curve.

• It provides an accessible alternative to more complex methods, although it might be less accurate over larger intervals.
• It is best used with smaller step sizes, as this increases accuracy by reducing the error that accumulates with each predictive step.

This method is particularly valuable because it gives insight into how functions behave over time, making it beneficial for simulations and predictions in scientific research.

Initial Value Problem

An initial value problem is a type of differential equation that comes with a specified initial condition. The solution of the differential equation is constrained to pass through a given point in the plane. For instance, the exercise initially provides \( y(0) = 2 \), meaning that when \( x = 0 \), \( y \) must equal 2. This initial condition helps to uniquely determine the solution to the differential equation.

The significance of initial value problems lies in their ability to model real-world scenarios where a system's state at an initial time can be used to predict its future behavior. By giving both the differential equation and the initial condition, the problem is well-defined, enabling a specific solution path to be generated using methods such as Euler's Method.

Step Size in Numerical Methods

The step size, denoted by \( h \) in Euler's Method, is a crucial parameter that influences the accuracy and efficiency of numerical approximations. It defines the interval between two consecutive points along the solution path.

A smaller step size results in more steps to cover the same interval, leading to higher accuracy at the cost of increased computational effort. Conversely, a larger step size accelerates computation but can introduce significant errors due to less precise approximations at each step.

• For the exercise, \( h = 0.2 \) was chosen to balance accuracy and computation time over the interval \([0, 1]\).
• The choice of step size is pivotal: too large a step size might overlook important changes, while too small a step size might make computations unnecessarily complex.

Careful selection of the step size is essential for achieving both efficient and reliable approximations when solving differential equations numerically.