Problem 16 - Calculus

Question

Consider the differential equation. $$\frac{d y}{d x}=\sin (x y), \quad \text { with initial condition } y(1)=1$$ Estimate \(y(2),\) using Euler's method with step sizes \(\Delta x=0.2,0.1,0.05 .\) Plot the computed approximations for \(y(2)\) against \(\Delta x .\) What do you conclude? Use your observations to estimate the exact value of \(y(2)\)

Step-by-Step Solution

Answer

As \( \Delta x \) decreases, the approximation stabilizes, indicating the estimated exact \( y(2) \).

Step 1: Understand Euler's Method

Euler's method is an iterative process to approximate solutions to differential equations of the form \( \frac{d y}{d x} = f(x, y) \) starting from an initial condition \( y(x_0) = y_0 \). Given \( \Delta x \), the next value is \( y_{n+1} = y_n + f(x_n, y_n) \Delta x \). Here, \( f(x, y) = \sin(xy) \).

Step 2: Initialize Values for Euler's Method

Start with \( x_0 = 1 \) and \( y_0 = 1 \). You need to compute \( y(2) \), which involves iterating from \( x_0 \) to \( 2 \) using steps of \( \Delta x = 0.2, 0.1, \) and \( 0.05 \).

Step 3: Apply Euler's Method for \( \Delta x = 0.2 \)

Using \( \Delta x = 0.2 \):- **Iteration 1:** Compute \( y_1 = y_0 + \sin(1 \cdot 1) \cdot 0.2 = 1 + 0.2 \sin(1) \).- **Iteration 2:** Update \( x_1 = 1.2 \), compute \( y_2 = y_1 + \sin(1.2 y_1) \cdot 0.2 \).- **Iteration 3:** Update \( x_2 = 1.4 \), compute \( y_3 = y_2 + \sin(1.4 y_2) \cdot 0.2 \).- **Iteration 4:** Update \( x_3 = 1.6 \), compute \( y_4 = y_3 + \sin(1.6 y_3) \cdot 0.2 \).- **Iteration 5:** Update \( x_4 = 1.8 \), compute \( y_5 = y_4 + \sin(1.8 y_4) \cdot 0.2 \).- **Iteration 6:** Final \( y_6 = y_5 + \sin(2 y_5) \cdot 0.2 \). The value is \( y(2) \) for \( \Delta x = 0.2 \).

Step 4: Apply Euler's Method for \( \Delta x = 0.1 \)

Repeat the process:- For \( \Delta x = 0.1 \), it takes 10 steps from \( x = 1 \) to \( x = 2 \).- Each step is: \( y_{n+1} = y_n + \sin(x_n y_n) \cdot 0.1 \).- Calculate until \( y(2) \) is obtained.

Step 5: Apply Euler's Method for \( \Delta x = 0.05 \)

Repeat again:- For \( \Delta x = 0.05 \), it takes 20 steps from \( x = 1 \) to \( x = 2 \).- Follow similarly: \( y_{n+1} = y_n + \sin(x_n y_n) \cdot 0.05 \).- Continue calculations until \( y(2) \) is found.

Step 6: Plotting the Results

For each \( \Delta x \), plot the value of \( y(2) \) on the y-axis against \( \Delta x \) on the x-axis. You should see that as \( \Delta x \) decreases, the approximation of \( y(2) \) stabilizes to a certain value.

Step 7: Estimate the Exact Value of \( y(2) \)

Observe the trend in the plot. As \( \Delta x \) approaches 0, the \( y(2) \) values converge. The closer \( x \) approaches a specific value, the better approximation it provides for the exact \( y(2) \). Conclude that as \( \Delta x \) decreases, \( y(2) \) seems to be approaching a certain value more consistently.

Key Concepts

Differential Equations

Differential equations are equations that involve an unknown function along with its derivatives. They play a vital role in modeling how different quantities change over time and space. A simple example of a differential equation is the one given in the exercise: \[\frac{d y}{d x} = \sin(x y)\]Here, \(\frac{d y}{d x}\) represents the rate of change of \(y\) with respect to \(x\), and this rate is given by the function \(\sin(xy)\). Differential equations are foundational for understanding phenomena in physics, engineering, biology, and finance. They help in predicting the future behavior of these systems based on certain initial conditions.

Numerical Approximation

Numerical approximation refers to finding an approximate solution to a problem that cannot easily be solved with analytical methods. Euler's method, as shown in our exercise, is a simple yet powerful technique used to approximate solutions of differential equations. It uses the idea of taking small steps to march forward along the curve defined by the equation.Euler's method approximates the next value \(y_{n+1}\) as follows:

• Start from an initial value \(y_0\).
• Use the derivative function \(f(x,y)\) to calculate the change \(\Delta y = f(x, y)\Delta x\).
• Move from \(y_n\) to \(y_{n+1} = y_n + \Delta y\).

The accuracy of this method improves as the step size \(\Delta x\) decreases, leading to a better approximation of the solution. This is particularly useful when dealing with complex equations or when an exact analytical solution is not feasible.

Initial Value Problem

An initial value problem (IVP) is a type of differential equation along with a specific starting condition that provides enough information for a unique solution. In the exercise, the initial value condition is given as \(y(1) = 1\). This tells us that when \(x = 1\), the value of \(y\) is 1. The importance of the initial condition lies in its ability to influence the entire behavior of the solution. Depending on this starting point, the solution to the differential equation can differ significantly. IVPs are crucial in modeling systems where the initial state is known, and the task is to predict the behavior over time or another variable. Using Euler's method, we start from this initial condition and incrementally build the solution through each small step. This is how we estimate future values, like \(y(2)\) in this exercise, by relying on the initial information provided.