Question

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

Short answer

The exact value of y at x=1.7 using Euler's Method with increments of \(\Delta x=-0.1\) depends on the calculations performed in Step 4. To get this value, you would need to perform the iterative calculations, updating the value of y at each step until reaching x=1.7.

Step-by-step solution

Understand Euler's Method

Euler's Method is a numerical method used to approximate solutions to first order differential equations with a given initial value. It involves taking small steps along the curve, calculating the slope at each step, and using this to estimate the next point.

Set Up Initial Conditions

The initial condition provided in the exercise is x=2, y=1. This is the starting point from which we will 'walk' towards x=1.7 in increments of Δx=-0.1. We also have the differential equation dy/dx=2-x

Apply Euler's Method

We apply Euler's Method iteratively to calculate the estimated values of y at each step, until we reach x=1.7. The formula used in each step is: y_{i+1}=y_i + Δx ( dy/dx evaluated at x=x_i). In our case, dy/dx=2-x.

Calculate

Let's begin the calculation. Starting with x=2, y=1, evaluate the differential equation (2-x), multiply it by Δx, and add it to the current y. Repeat these steps until you reach x=1.7. Here is the step-by-step process:1) For \(x_1 = 2, y_1 = 1\), \(f(x_1, y_1) = 2 - x_1 = 0\). So, \(y_2 = y_1 + Δx * f(x_1, y_1) = 1 - 0.1*0 = 1\).2) For \(x_2 = 2 - 0.1 = 1.9, y_2 = 1\), \(f(x_2, y_2) = 2 - x_2 = 0.1\). So, \(y_3 = y_2 + Δx * f(x_2, y_2) = 1 - 0.1*0.1 = 0.99\).3) You keep iterating in this way until you reach \(x=1.7\)

Final Answer

The process should continue until reaching x=1.7. To facilitate the calculation it might be helpful to use a spreadsheet or a computational software. The result obtained at x=1.7 would be the approximate solution to the differential equation at that point using Euler's method with increments of \(\Delta x=-0.1\)

Key concepts

Differential Equations

Differential equations are powerful tools used to describe the relationships between varying quantities and their rates of change. They come in many forms, often reflecting real-world phenomena such as population growth, chemical reactions, and even the motion of celestial bodies. In essence, a differential equation relates a function with one or more of its derivatives.

At the core of our exercise is a first-order differential equation given by \( \frac{d y}{d x}=2-x \). This equation specifies that the rate at which \( y \) changes with respect to \( x \) is related linearly to the value of \( x \). This type of equation can model simple phenomena but still provides interesting solutions to explore using numerical methods like Euler's Method, especially when finding an exact solution is complex or impossible.

Numerical Methods

When exact solutions to differential equations are not available, or when we prefer a quick approximation, numerical methods come into play. These techniques provide a way to estimate solutions to mathematical problems using iterative calculations. The beauty of such methods lies in their ability to tackle complex problems that defy analytical solutions, making them invaluable in fields such as engineering, physics, and finance.

Euler's Method, which is the focus of our exercise, is one of the simplest numerical methods. This method approximates the solution by taking small steps along the function's domain and calculating the function's slope at each step. By doing so iteratively, we build a staircase that follows the curve of the solution, getting close to the actual path the function would take. The result is a series of points that provide an approximation of the function's graph.

Initial Value Problems

Initial value problems (IVPs) are a class of differential equations specifying not just the relationship between variables and their derivatives, but also an initial point to start from. This point, given by specific values for \( x \) and \( y \) coordinates, serves as a springboard for the numerical method to begin its approximations.

The problem assigned involves an IVP where we know that when \( x=2 \) the solution \( y \) is 1. The goal is to determine the value of \( y \) when \( x=1.7 \) using Euler's Method with increments of \( \Delta x=-0.1 \). This method hinges on using the initial point to predict the value of \( y \) at the next \( x \) by adding the product of \( \Delta x \) and the slope obtained from the differential equation to the current value of \( y \) at each step. We then observe how \( y \) evolves as we work backwards in \( x \), getting progressively closer to the sought-after value at \( x=1.7 \) through each calculated step.