Question

Use the Euler method to approximate the values of the exact solution to the initial-value problem $$\begin{gathered} y^{\prime}=\frac{\sin x-y}{\cos y+x} \\\y\left(\frac{\pi}{2}\right)=\pi\end{gathered}$$ at \(x_{n}=\pi / 2+n h\), for \(n=1, \ldots, 5\), and \(h=\pi / 10 .\) Find the exact solution. Can it be solved readily for \(y\) terms of \(x ?\)

Short answer

We used the Euler method to approximate the solution of the initial-value problem (IVP) with the given ODE \(y' = \frac{\sin x - y}{\cos y + x}\) and initial condition \(y\left(\frac{\pi}{2}\right) = \pi\). The approximations were found at the points \(x_n = \frac{\pi}{2} + n\frac{\pi}{10}\) for \(n=1,...,5\) using a step size \(h = \frac{\pi}{10}\). However, finding an exact solution to this non-linear ODE is not readily possible as it doesn't have a separable form, and no obvious analytical method seems applicable.

Step-by-step solution

Write down the given IVP

The given IVP can be written as: \[y' = \frac{\sin x - y}{\cos y + x}\] with the initial condition \(y\left(\frac{\pi}{2}\right) = \pi\).

Apply the Euler method

The Euler method is a numerical technique to approximate the solution of an IVP. The recursive formula for the Euler method is: \[y_{n+1} = y_n + hf(x_n, y_n)\] where n is the step number, \(h\) is the step size, and \(f(x_n,y_n)\) is the given ODE function. In our case, \(h = \frac{\pi}{10}\) and \(f(x, y) = \frac{\sin x - y}{\cos y + x}\).

Compute the approximations for the given points

We want to find the values at \(x_n = \frac{\pi}{2} + n\frac{\pi}{10}\) for \(n=1,...,5\). We will compute the Euler approximations for these points iteratively using the recursive formula in Step 2: 1. For n=1: \(x_1 = \frac{\pi}{2} + \frac{\pi}{10}\) - \(y_1 = y_0 + h\cdot f(x_0, y_0)\) - \(y_1 = \pi + \frac{\pi}{10}\cdot \frac{\sin(\frac{\pi}{2}) - \pi}{\cos(\pi) + \frac{\pi}{2}}\) 2. For n=2: \(x_2 = \frac{\pi}{2} + 2\frac{\pi}{10}\) - \(y_2 = y_1 + h \cdot f(x_1, y_1)\) - Compute \(f(x_1, y_1)\) using the value of \(y_1\) from the previous step, then update \(y_2\). 3. Repeat this process for \(n=3,4,5\).

Find the exact solution

We are asked if it is possible to find an exact solution to the given ODE in terms of y and x. Upon inspection, we can see that the ODE is non-linear and does not have a separable form. Additionally, there is no obvious substitution or method to solve this ODE analytically. Therefore, it seems unlikely that an exact solution can be found readily. In conclusion, we have computed Euler approximations for the given points as per the problem statement and have determined that finding an exact solution for this ODE is not straightforward, if even possible.

Key concepts

Ordinary Differential Equations

An ordinary differential equation (ODE) involves functions of one independent variable and their derivatives. These equations express relationships between a function and its rate of change. In the given exercise, the differential equation is \(y' = \frac{\sin x - y}{\cos y + x}\). Here, \(y\) is a function of \(x\), and \(y'\) represents the rate of change of \(y\) with respect to \(x\).
This ODE is nonlinear due to the presence of variables \(y\) and \(x\) in both the numerator and the denominator, making it complex to solve using standard analytical methods.
Understanding ODEs is fundamental as they appear in various fields such as physics, engineering, and economics, where they model real-world phenomena like motion, growth, and decay.

Initial Value Problem

An initial value problem (IVP) in differential equations is where the solution to the ODE is sought, provided the values of the function at a specific point. This extra information helps to uniquely determine the solution.
In our exercise, the initial condition is \(y\left(\frac{\pi}{2}\right) = \pi\). This specifies the value of \(y\) when \(x = \frac{\pi}{2}\).
By using this initial value, we can use numerical methods like Euler’s method to find successive approximations of \(y\) at other points along \(x\). Initial value problems are essential in physics for predicting future states of a system given its current state.

Numerical Methods

Numerical methods provide techniques to approximate solutions to mathematical problems that cannot be solved analytically. Euler's method is one such approach for solving ODEs, particularly useful in initial value problems.
The method uses the formula \(y_{n+1} = y_n + h f(x_n, y_n)\) to compute the approximation step by step. Here, \(h\) is the step size, \(x_n\) and \(y_n\) are the current values, and \(f(x_n, y_n)\) is the function defined by the ODE.
In the exercise, a step size \(h = \frac{\pi}{10}\) is used, providing a balance between accuracy and computational effort. Euler's method is straightforward but can accumulate errors if a smaller step size is not chosen carefully.

• Advantage: Simple and easy to implement.
• Limitation: Can be inaccurate for stiff equations or large interval steps.

Nonlinear Differential Equations

Nonlinear differential equations involve terms that are nonlinear in the unknown function or its derivatives, making them more complicated than linear ones. In the given problem, the equation is nonlinear due to the mixed terms of \(x\) and \(y\).
These types of equations often do not have straightforward solutions.
Nonlinear ODEs can model complex, real-world systems like predator-prey dynamics, electrical circuits, or chaotic systems. Finding exact solutions is usually difficult or impossible, prompting the use of numerical methods to approximate solutions.

• Characteristics: Complex, sensitive to initial conditions, and often exhibit behavior not seen in linear counterparts.
• Approaches: Use numerical methods like Euler’s or Runge-Kutta for approximations, or explore graphical methods.