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 \(h .\) $$ y^{\prime}=\cos x+\sin y, \quad y(0)=5, \quad n=10, \quad h=0.1 $$

Short answer

As the exact output values vary, it's not feasible to provide a concrete answer here. Typically, you'll observe that these estimated values of y deviate from the exact values by a certain amount, illustrating the inherent error in numerical methods like Euler's.

Step-by-step solution

Initial Setup

First, identify and write down all the given values and the differential equation to be solved. The differential equation is \(y' = \cos x + \sin y\), the initial value of y is \(y(0) = 5\), the number of steps n is 10 and the step size h is 0.1.

Expression for Euler's Method

The iterative formula for Euler's method is \(y_{i+1} = y_i + hf(x_i, y_i)\). Here, the function \(f(x,y) = \cos x + \sin y\).

Apply Euler's Method Iteratively

Apply the iterative formula to estimate the numeric solution of the differential equation. Start with \(x_0 = 0\) and \(y_0 = 5\) and apply the formula above.

Tabulate Results

After calculating the values of y for each step from \(i = 0\) to \(i = 10\), summarize the results in a table. Each row of the table should include the value of i, the corresponding values of x and y.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They have a crucial role in describing real-world phenomena where change occurs. These equations depict how a particular quantity evolves over time or space. For instance, how populations grow, how heat diffuses across objects, or how velocities change. Differential equations are often classified into ordinary and partial types. Ordinary differential equations (ODEs) involve functions of a single variable and its derivatives, while partial differential equations (PDEs) involve functions of multiple variables.

In our exercise, we have an ordinary differential equation given by \( y' = \cos x + \sin y \). This equation implies how the derivative of \( y \) depends on \( x \) and \( y \) itself. Solving a differential equation like this often involves finding a function \( y(x) \) that satisfies the equation for a range of \( x \).

However, analytical solutions can be challenging to derive for many differential equations, especially when they include non-linear components, such as \( \sin y \). This is where numerical methods, like Euler's Method, become highly useful.

Numerical Methods

Numerical methods offer strategies to approximate solutions to mathematical problems that might not have straightforward exact answers. When it comes to solving differential equations, numerical methods can help us estimate the function instead of deriving it exactly. These methods break down the problem into smaller pieces, which can be solved with simple arithmetic.

Euler's Method is one of the basic numerical techniques to solve initial value problems of ordinary differential equations. Here’s how it works:

• Begin at a known initial value.
• Proceed in small steps along the line, updating your estimate of the solution by adding a small increment based on the function's slope.

As noted in the exercise, you use an iterative formula for Euler's Method: \( y_{i+1} = y_i + hf(x_i, y_i) \). Here \( f(x, y) \) is \( \cos x + \sin y \), and \( h \) is the step size.

Despite its simplicity, Euler's Method is a powerful tool. It provides a sequence of estimates, which can give insights into the solution's behavior over the interval of interest.

Initial Value Problems

Initial value problems are a type of differential equation problem where the solution is determined from a specified initial condition. These kinds of problems are a fundamental aspect in modeling dynamic systems where initial conditions greatly influence the outcome.

For the exercise, the initial value is given as \( y(0) = 5 \). This indicates the starting point of the function \( y(x) \) at \( x = 0 \). From here, Euler's Method allows us to proceed from this known condition to estimate the values of \( y \) at later points, given the function's provided derivative.

In real-world applications, setting the correct initial conditions is crucial as small changes can lead to vastly different outcomes. This sensitivity to initial conditions is seen in many fields, like physics, engineering, and economics.

In summary, tackling initial value problems with numerical methods such as Euler's Method helps to approximate solutions where analytical methods might be intractable, expanding our ability to model and understand complex systems.