Question

In Exercises \(45-48\) , use Euler's Method with increments of \(\Delta x=-0.1\) to approximate the value of \(y\) when \(x=1.7\) \(\frac{d y}{d x}=1+y\) and \(y=0\) when \(x=2\)

Short answer

The implementation of Euler's method as detailed above will provide an approximation to \(y\) when \(x=1.7\). The exact value cannot be determined without implementing these steps.

Step-by-step solution

Understanding Euler's Method

Euler's Method is a simple and powerful numerical method for solving ordinary differential equations with a given initial value. The general formula for Euler's method is \(y_{i+1} = y_{i} + f(x_{i}, y_{i})\Delta x\). Here, \(\frac{dy}{dx} = 1 + y\) will be our function \(f(x_{i}, y_{i})\)

Set Initial Values

The exercise has given us that \(y=0\) when \(x=2\). This is our initial condition, and we will start our calculations from this point. So, \(x_{0} = 2\) and \(y_{0} = 0\).

Compute Next Values Using Euler's Method

Using Euler's method to compute the next approximated values, \(y_{i+1} = y_{i} + (1 + y_{i})\Delta x\). The increments have already been defined as \(\Delta x = -0.1\). We will continue this operation until \(x=1.7\).

Iterate Until Reaching Final Value of X

We keep applying Euler's method and iterate until the \(x\) value is equal to 1.7.

Calculate Final Approximated Value of Y

After we have the last \(y\) value, this will be the approximation of \(y\) when \(x=1.7\) via Euler's Method

Key concepts

Ordinary Differential Equations

Ordinary differential equations (ODEs) are equations involving derivatives that describe how a certain quantity changes over time or another variable. For example, the equation from the exercise, \(\frac{d y}{d x}=1+y\), is an ODE that models the rate at which \(y\) changes with respect to \(x\). ODEs are fundamental in expressing physical laws, population dynamics, financial models, and many other phenomena.

In our specific exercise, the ODE gives us the relationship between \(x\) and \(y\) that, when solved, would yield a function \(y(x)\) illustrating how \(y\) behaves as \(x\) changes. Finding an exact solution can be complex, or sometimes impossible, which is where numerical methods such as Euler's Method come into play.

Numerical Methods

Numerical methods are techniques employed to obtain approximations to the solutions of mathematical problems, which can be too difficult or even impossible to solve analytically. Euler's Method, used in our problem, is one of the simplest numerical methods for solving ODEs. It is an iterative process that starts with an initial value and then progresses step by step based on a chosen increment \(\Delta x\).

The beauty lies in its simplicity: The current value of \(y\) is used to estimate the slope of the tangent to the curve at that point, which is then used to estimate the next value of \(y\). Despite its simplicity, Euler's Method can be quite powerful, especially when the step size is small enough to increase the accuracy of the approximation.

Initial Value Problems

An initial value problem is a type of differential equation accompanied by a condition that specifies the value of the unknown function at a given point, normally referred to as the initial condition. In the exercise, the initial condition is \(y=0\) when \(x=2\), which means we know where to start our process of finding \(y\) for other values of \(x\).

Initial value problems are crucial because they ensure the uniqueness of the solution to an ODE. Without an initial value, there could be infinitely many solutions to the same differential equation. By implementing Euler's Method, we use the given initial condition to approximate the solution, step by step, obtaining \(y\) values at successive points along the domain of \(x\), ultimately helping us to build a piecewise linear approximation to the original curve defined by the ODE.