Problem 1 - Advanced Engineering Mathematics

Question

Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of (3) by hand, first using \(h=0.1\) and then using \(h=0.05\). $$ y^{\prime}=2 x-3 y+1, \quad y(1)=5 ; y(1.2) $$

Step-by-Step Solution

Answer

With steps h=0.1 and h=0.05, the approximations for y(1.2) are 3.07 and 2.9675 respectively.

Step 1: Understand Euler's Method

Euler's method is a numerical procedure for approximating solutions to differential equations. Given a first-order differential equation \( y' = f(x, y) \), with an initial condition \( y(x_0) = y_0 \), the method uses the formula \( y_{n+1} = y_n + hf(x_n, y_n) \) to estimate \( y \) values at subsequent points. The parameter \( h \) is the step size.

Step 2: Set Up for Euler's Method with h=0.1

For the differential equation \( y' = 2x - 3y + 1 \), calculate the next \( y \) values starting from \( (x_0, y_0) = (1, 5) \) with \( h = 0.1 \). For the first iteration: \( f(x_0, y_0) = 2(1) - 3(5) + 1 = -12. \) Thus, \( y_1 = y_0 + h f(x_0, y_0) = 5 + 0.1(-12) = 3.8. \) Find \( y(1.2) \) by repeating the process until \( x = 1.2 \).

Step 3: Calculate Subsequent Points for h=0.1

Recalculate for additional steps: - \( x_1 = 1.1, y_1 = 3.8 \) with \( f(x_1, y_1) = 2(1.1) - 3(3.8) + 1 = -7.3 \). - Find \( y_2 = y_1 + 0.1(-7.3) = 3.07. \)- \( x_2 = 1.2 \), checking the change before 1.2.

Step 4: Conclusion for h=0.1

At \( x = 1.2 \), after the final adjustment, the approximation is \( y(1.2) \approx 3.07 \).

Step 5: Set Up for Euler's Method with h=0.05

Repeat the process with \( h = 0.05 \). Recall initial point is \( (x_0, y_0) = (1, 5) \). Calculate the first step with \( f(x_0, y_0) = -12 \), so \( y_1 = 5 + 0.05(-12) = 4.4. \)

Step 6: Calculate Subsequent Points for h=0.05

Continue computing: - For \( x_1 = 1.05, y_1 = 4.4 \), find \( f(x_1, y_1) = 2(1.05) - 3(4.4) + 1 = -8.15 \).- Then \( y_2 = 4.4 + 0.05(-8.15) = 3.9925 \). - Continue this calculation until \( x = 1.2 \).

Step 7: Conclusion for h=0.05

At \( x = 1.2 \), the calculated approximation is \( y(1.2) \approx 2.9675 \) after all intermediate \( y \) values are calculated with the refined step size.

Key Concepts

Numerical Procedures

Numerical procedures are methods used to find approximate solutions to mathematical problems when exact solutions are not easily obtainable. In the context of differential equations, these procedures provide a way to approximate the value of the unknown function at different points. One common numerical method is Euler's method.

• It is particularly useful for solving ordinary differential equations, where traditional analytical methods might be complex or impossible to implement.
• These numerical techniques can provide good approximations rapidly, making them suitable for computer algorithms.
• By relying on iterative calculations, these methods progressively approximate the solution from an initial condition, making the process more manageable.

Euler's method, for instance, employs simple arithmetic operations repeatedly to estimate the solution over an interval. Its main advantage lies in its simplicity and ease of implementation, especially for educational purposes.

Differential Equations

Differential equations are equations that relate functions with their derivatives, indicating how a function changes over time or space. They are essential in various fields such as physics, engineering, and economics to model dynamic systems.

• The primary goal of solving a differential equation is to find a function that satisfies the relationship specified by the equation.
• In an ordinary differential equation (ODE), the function depends on a single variable.
• Euler's method is often utilized to approximate solutions for first-order ODEs, where the solution or the unknown function and its derivative are involved.

For example, the differential equation given in the exercise, \( y' = 2x - 3y + 1 \), involves variables both \( x \) and \( y \). The equation describes how \( y \) changes with respect to \( x \), and Euler's method helps us predict \( y \) values at specific points along this curve.

Step Size in Numerical Methods

The step size, denoted by \( h \), plays a crucial role in numerical methods like Euler's method. It represents the distance between consecutive points where the solution is approximated.

• A smaller step size generally increases the accuracy of the approximation. This is because the method can capture more details of the function's behavior between points.
• Conversely, a larger step size can result in a faster computation but may lead to less accurate results.
• Choosing an appropriate step size is a trade-off between computational efficiency and accuracy.

In the exercise, two different step sizes, \( h = 0.1 \) and \( h = 0.05 \), are tested.
The smaller step size \( h = 0.05 \) provided a more accurate approximation at \( x = 1.2 \), as it captures the function's changes more precisely than a larger step size \( h = 0.1 \). This example illustrates why careful consideration of the step size is essential in numerical procedures to achieve a balance between speed and precision.