Question

Euler's Method In Exercises \(73-78\) , use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use \(n\) steps of size \(h .\) $$ y^{\prime}=x+y, \quad y(0)=2, \quad n=10, \quad h=0.1 $$

Short answer

The final solution is an approximate value of \(y\) after \(n\) steps using Euler's Method based on the given initial condition, differential equation, and step size.

Step-by-step solution

Identify the given values

The given differential equation is \(y'=x+y\). The initial condition is \(y(0)=2\). The step size, \(h\), is 0.1 and the total number of steps, \(n\), is 10.

Initialize the values

Start with \(x_{0}=0\), \(y_{0}=2\), \(h=0.1\), and \(n=10\). These values are based on the given initial condition and specified step size and number of steps.

Apply Euler's Method

Apply Euler's Method formula \(y_{i+1}=y_{i}+hf(x_{i},y_{i})\), here \(f(x,y)=x+y\), for \(i=0\) to \(9\). For the first step, with \(i=0\), the left-hand side will be \(y_{1}\) and the right-hand side will be \(y_{0}+0.1f(x_{0},y_{0})\). Subsitute \(y_{0}\) and \(x_{0}\) and compute \(y_{1}\). Repeat steps for \(i=1\) to \(9\). For each \(i\), update \(x_{i+1}=x_{i}+h\) and compute \(y_{i+1}\) using the formula.

Present the results

After completing the steps, the table of approximate solution values of \(y\) as per the Euler's method will be completed. The \(y_{10}\) value is the final approximation.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They are crucial in expressing the rates of change and can describe a wide range of phenomena like population dynamics, physics, and engineering problems. The general form of a first-order differential equation, which we use here, is given by: \( y' = f(x, y) \).
In this context, \( y' \) represents the rate of change of \( y \) with respect to \( x \). Essentially, it tells us how \( y \) changes as \( x \) changes. In the exercise problem, the differential equation \( y' = x+y \) helps us understand how \( y \) evolves depending on both its current value and the input \( x \).
Understanding differential equations is fundamental as they provide a way to predict future behavior based on current or past conditions. Euler's Method, used here, is a numerical approach to finding approximate solutions to these equations, especially when an exact analytical solution may be difficult or impossible to obtain.

Initial Value Problem

An initial value problem is a type of differential equation that comes with an initial condition. This condition specifies the value of the function at a particular point, which is essential for determining a unique solution to the equation. In differential equations, an initial value problem typically has the form: \( y(a) = b \).
Here, "\( y(0)=2 \)" is our initial condition, meaning at \( x=0 \), the function \( y \) is equal to 2. This is a starting point for our calculations when employing Euler's Method. The initial value sets the stage for how you will compute the subsequent values in the table.
Understanding initial value problems is critical as they ensure that the numerical methods, such as Euler’s Method, have something concrete to begin with, allowing for the construction of a predictable path for solving the differential equation over a defined range.

Numerical Methods

Numerical methods are techniques used to approximate solutions for mathematical problems that may not have an exact solution. They are particularly useful in complex differential equations where analytical solutions are difficult. Euler’s Method is one such numerical method.
Euler’s Method approximates the solution of a differential equation by taking small incremental steps from an initial value, calculating the slope at each step, and using this slope to update the value of \( y \). It leverages the formula: \[ y_{i+1} = y_i + h \cdot f(x_i, y_i) \]
Here, \( y_i \) and \( x_i \) are current values, \( h \) is the step size, and \( f(x_i, y_i) \) gives the slope. It's a straightforward method and often used as a first approach before more sophisticated methods.
Overall, numerical methods like Euler’s provide practical tools for real-world problems where precision is important, and the groundwork for understanding more advanced techniques.

Step Size

Step size, denoted as \( h \), plays a crucial role in numerical methods like Euler's Method. It determines how big the increments are when moving from one calculated point to the next. A smaller \( h \) means more steps are taken, potentially increasing the accuracy of approximation. However, it comes at the cost of increased computation.
In the exercise, the step size is given as 0.1. This means every successive \( x \) value is computed as \( x_{i+1} = x_i + 0.1 \). The decision for this \( h \) is a balance between precision and computation effort. Smaller values provide finer approximations, but require more work. Larger values may speed up the process but can overshoot and lead to less accurate solutions.
Choosing an appropriate step size requires understanding the problem's demands and the acceptable trade-off between precision and computational efficiency. It's a critical part of applying Euler’s Method effectively.