Problem 37 - Calculus Concepts

Question

Use Euler's method and two steps to estimate \(y\) when \(x=8\), given \(\frac{d y}{d x}=\frac{5}{x}\) with initial condition \((2,2) .\)

Step-by-Step Solution

Answer

Using Euler's method, the estimated value of \( y \) at \( x = 8 \) is 12.5.

Step 1: Identify Initial Conditions and Step Size

We are given the differential equation \( \frac{dy}{dx} = \frac{5}{x} \) with the initial condition \((x_0, y_0) = (2, 2) \). We need to find the value of \( y \) at \( x = 8 \) using Euler's method with two equal steps. Since we are moving from \( x_0 = 2 \) to \( x = 8 \), and there are two steps in total, the step size \( h \) can be calculated as follows: \[ h = \frac{8 - 2}{2} = 3. \]

Step 2: Apply Euler's Method for First Step

Starting from the initial point \((x_0, y_0) = (2, 2)\), the next \(x\)-value is calculated using \(x_1 = x_0 + h\) which gives \( x_1 = 2 + 3 = 5 \). The next \(y\)-value is estimated using the formula \(y_1 = y_0 + h \cdot f(x_0, y_0)\), where \(f(x, y) = \frac{5}{x}\). Substituting the values, we get: \[ y_1 = 2 + 3 \times \frac{5}{2} = 2 + 7.5 = 9.5. \] Thus, the estimated point after the first step is \((5, 9.5)\).

Step 3: Apply Euler's Method for Second Step

Using the new point \((x_1, y_1) = (5, 9.5)\), calculate \( x_2 = x_1 + h = 5 + 3 = 8 \). Now, estimate the next \(y\)-value using \(y_2 = y_1 + h \cdot f(x_1, y_1)\). Substitute the values to get: \[ y_2 = 9.5 + 3 \times \frac{5}{5} = 9.5 + 3 = 12.5. \] Thus, the estimated value of \( y \) at \( x = 8 \) is \( y_2 = 12.5 \).

Key Concepts

Euler's Method

Euler's Method is a simple numerical technique for solving ordinary differential equations (ODEs). It provides an approximate solution to an initial value problem by using a step-by-step approach. The core idea is to use the slope of the tangent line on a small interval to estimate the new value of the dependent variable.
Imagine it as sketching the curve of the solution by plotting small, linear segments.

• Start with an initial value, known as the initial condition, which specifies one point on the solution curve (e.g., \(x_0, y_0\)).
• Compute the slope of the function using the given differential equation \(\frac{dy}{dx}\).
• Use this slope to predict the next value of the function after a small step size.
• Repeat the process to continue estimating more points along the curve.

Differential Equations

Differential equations are mathematical equations relating a function with its derivatives. They play a crucial role in describing various phenomena in science and engineering, such as heat conduction, wave propagation, and population dynamics.
In the given exercise, \(\frac{dy}{dx} = \frac{5}{x}\) is a differential equation expressing the rate of change of \(y\) with respect to \(x\). The equation is telling us how \(y\) changes as \(x\) changes.
To solve a differential equation means to find a function \(y\) that satisfies the equation for a range of values. In many cases, especially when an equation is complex or lacks an analytical solution, numerical methods like Euler's can be employed to approximate the solution.

Initial Value Problem

An initial value problem (IVP) is a type of differential equation problem that seeks to find a function satisfying a differential equation along with initial conditions. These initial conditions provide specific values for the function at a particular point, allowing for a specific, unique solution.
In our example, the initial value problem is given by the equation \(\frac{dy}{dx} = \frac{5}{x}\) with the condition \(x_0 = 2\), \(y_0 = 2\). This means, when \(x = 2\), then \(y = 2\). These initial conditions are critical as they anchor the solution to a particular starting point, from which we use Euler's method to explore how the function behaves as \(x\) increases.

Step Size Calculation

Step size is a crucial element in numerical methods like Euler's method. It determines the interval at which function values are approximated and subsequently joined together to sketch the solution curve.
For our exercise, we need to calculate the step size \(h\) based on the number of steps we wish to take between the initial point and our point of interest. Here, we need to estimate \(y\) at \(x = 8\) starting from \(x_0 = 2\) with two equal steps. Hence, the step size is calculated as \[h = \frac{8 - 2}{2} = 3.\]
Choosing the appropriate step size

• A smaller step size generally yields a more accurate approximation but takes more computations.
• A larger step size reduces computation but can lead to significant errors, especially if the function has high curvature.

In practical terms, step size plays a balancing act between computational efficiency and accuracy.