Question

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

Short answer

The step-by-step solution will provide the approximate value of \(y\) at \(x=1.7\) by Euler's method.

Step-by-step solution

Specify the function based on the differential equation

First express the function of the differential equation given as \(f(x, y) = x - y\)

Applying Euler's method formula

Following the Euler's method, \(y_{i+1} = y_i + \Delta x * f(x_i, y_i)\), where \(\Delta x=-0.1\), and the starting point is \(x_i=2\), \(y_i=2\)

Calculation for each step until \(x = 1.7\)

Calculate \(y\) value for each step. Start with \(x=2\) and \(y=2\). Substitute these values into Euler's formula until you reach \(x = 1.7\).

Interpret the result

The last calculated \(y\) value will be the approximate solution needed for \(y\) when \(x=1.7\).

Key concepts

Differential Equations

Differential equations are mathematical equations that describe how a particular quantity changes over time, depending on other variables. In the provided example, the differential equation is given by \(\frac{d y}{d x}=x-y\), which expresses the rate of change of a function \(y\) with regard to another variable \(x\). These types of equations are pivotal in fields such as physics, engineering, economics, and biology because they can model a wide variety of phenomena, from the motion of planets to the spread of diseases.

The equation from the exercise is a first-order differential equation because it contains the first derivative of the function \(y\). Solving differential equations often involves finding a function that satisfies the given relationship between its derivatives and other functions or constants. In some cases, an explicit solution can be obtained algebraically, but many differential equations don't have a straightforward solution and that's where numerical approximation methods like Euler's Method come into play.

Numerical Approximation

Numerical approximation is a branch of numerical analysis that deals with finding approximate solutions to mathematical problems that might be difficult or impossible to solve exactly. Euler's method, as seen in the exercise, is a fundamental numerical technique used to approximate solutions to differential equations.

When we apply numerical techniques, we settle for an estimated answer rather than an exact one. This approach is particularly useful when the equations are too complex for analytical solutions or where an exact formula is not available. Euler's method uses a step-by-step progression to estimate the function's values at certain points, making it a good option for solving initial value problems when we need to know the behavior of a system over a specific interval.

Initial Value Problem

An initial value problem is a type of differential equation along with a specific condition given at a starting point. This condition is known as the 'initial value'. In our context, the initial value problem is defined by the differential equation \(\frac{d y}{d x}=x-y\) with the initial condition \(y(2)=2\).

This starting point is critical because it sets the state of the system from which we will calculate all future states using the numerical method. The initial values serve as the launching pad, providing a concrete spot from which to project the behavior of the equation. Without an initial value, the problem would be indeterminate since differential equations can have infinitely many solutions.

Incremental Step Size

Incremental step size, denoted as \(\Delta x\) in numerical methods like Euler's Method, refers to the size of each step used in approximating the solution to a differential equation. In the exercise, an incremental step size of \(\Delta x=-0.1\) is chosen, which indicates that each step used to approach the solution from \(x=2\) to \(x=1.7\) decreases by 0.1 units.

The choice of step size can significantly impact the accuracy and efficiency of the numerical approximation. A smaller step size can lead to a more accurate solution but requires more computational work. Conversely, a larger step size might decrease the time needed for computation but at the cost of accuracy. Therefore, selecting an appropriate step size is a crucial aspect of the problem-solving process when using Euler's Method or similar numerical approaches.