Problem 16 - Calculus: Early Transcendental Functions

Question

Use Euler's method with \(h=0.1\) and \(h=0.05\) to approximate \(y(1)\) and \(y(2) .\) Show the first two steps by hand. $$y^{\prime}=x / y^{2}, y(0)=2$$

Step-by-Step Solution

Answer

Using Euler's method with \(h=0.1\) or \(h=0.05\) to find the values of \(y(1)\) and \(y(2)\) in the given differential equation, we find that both \(y(1)\) and \(y(2)\) equal 2 regardless of the chosen step size.

Step 1: Understand Euler's method

Euler's method is a simple method to solve first-order differential equations. The formula for Euler's method is \(y_{i+1} = y_{i} + h*f(x_i, y_i)\) where \(h\) is the step size, \(f\) is the derivative of \(y\) as a function of \(x\) and \(y\), and \(y_i\) and \(y_{i+1}\) are the \(y\) values at step \(i\) and step \(i+1\), respectively.

Step 2: Apply Euler's method for h=0.1

In this part, we will first apply the Euler's method with \(h = 0.1\) and then for \(h = 0.05\). Using the given info, we start with \(x_0 = 0, y_0 = 2\), and \(f(x_0, y_0) = 0/2^2 = 0\). Using the formula for Euler's method, we get \(y_1 = y_0 + h*f(x_0, y_0) = 2 + 0.1*0 = 2\). Following the same method again and calculating for \(y_2\), we find \(y_2\) equals 2.

Step 3: Apply Euler's method for h=0.05

Proceeding with \(h = 0.05\), again using the given info, we start with \(x_0 = 0, y_0 = 2\), and \(f(x_0, y_0) = 0/2^2 = 0\). Using the formula for Euler's method, we get \(y_1 = y_0 + h*f(x_0, y_0) = 2 + 0.05*0 = 2\). Calculating for \(y_2\), we find \(y_2\) equals 2.

Step 4: Interpret the results

Since both for \(h=0.1\) and \(h=0.05\), the results of \(y_1\) and \(y_2\) equal 2, this means that using Euler's method and for the given differential equation, \(y(1)\) and \(y(2)\) approximate to 2, irrespective of the step size.

Key Concepts

Differential Equations

Differential equations are equations that involve a function and its derivatives. They are fundamental in describing various natural phenomena, like how populations change over time, how heat distributes in a body, or how financial investments grow. The key idea is that they capture the relationship between an unknown function and its rate of change.
For example, in this exercise, the differential equation is given by:

• \( y' = \frac{x}{y^2} \)
• With an initial condition \( y(0) = 2 \)

This setup usually means we want to find a function \( y(x) \) that satisfies this equation and initial condition. These equations are vital across many fields of science and engineering as they model dynamic systems.

Numerical Approximation

Numerical approximation refers to the methods used to find approximate solutions to mathematical problems that may not have exact solutions easily available. When dealing with differential equations, exact solutions might be complex or impossible to find.
Euler's method is one of the simplest numerical approximation techniques. It helps estimate solutions to differential equations by approximating the next value of the function based on the current value.

• We start with an initial value, \( y_0 \), and iterate using the equation:
• \( y_{i+1} = y_{i} + h \cdot f(x_i, y_i) \)

This method provides a step-by-step solution by advancing a small increment or step size at a time. Although simple, it gives insight into how the solution behaves, making it useful in situations where a quick and approximate solution is needed.

Step Size in Numerical Methods

The step size, denoted by \( h \), plays a crucial role in the accuracy and efficiency of numerical methods like Euler's method. It is the increment in the independent variable \( x \) used to estimate the next value of the function.
The choice of step size affects the results:

• Smaller \( h \) generally increases accuracy but requires more computations.
• Larger \( h \) may lead to faster computations but can decrease the accuracy.

In this exercise, both \( h = 0.1 \) and \( h = 0.05 \) yielded the same result for \( y_1 \) and \( y_2 \) as 2. This happens because the derivative \( \frac{x}{y^2} \) is zero at \( x = 0 \), keeping the function value unchanged initially. However, in different cases, the choice of step size can significantly impact the result, highlighting its importance in achieving reliable approximations.