Problem 1 - Advanced Engineering Mathematics

Question

We identify \(f(x, y)=2 x-3 y+1 .\) Then, for \(h=0.1\) $$y_{n+1}=y_{n}+0.1\left(2 x_{n}-3 y_{n}+1\right)=0.2 x_{n}+0.7 y_{n}+0.1$$ and $$\begin{aligned} &y(1.1) \approx y_{1}=0.2(1)+0.7(5)+0.1=3.8\\\ &y(1.2) \approx y_{2}=0.2(1.1)+0.7(3.8)+0.1=2.98 \end{aligned}$$ For \(h=0.05\) $$y_{n+1}=y_{n}+0.05\left(2 x_{n}-3 y_{n}+1\right)=0.1 x_{n}+0.85 y_{n}+0.1$$ and $$\begin{aligned} y(1.05) & \approx y_{1}=0.1(1)+0.85(5)+0.1=4.4 \\ y(1.1) & \approx y_{2}=0.1(1.05)+0.85(4.4)+0.1=3.895 \\ y(1.15) & \approx y_{3}=0.1(1.1)+0.85(3.895)+0.1=3.47075 \\ y(1.2) & \approx y_{4}=0.1(1.15)+0.85(3.47075)+0.1=3.11514 \end{aligned}$$

Step-by-Step Solution

Answer

For \( h = 0.1 \), \( y(1.2) = 2.98 \). For \( h = 0.05 \), \( y(1.2) = 3.11514 \).

Step 1: Understanding the Function

We start by identifying the function given in the problem as \( f(x, y) = 2x - 3y + 1 \). This function will guide us in updating the values of \( y \) in each iteration.

Step 2: Setting Up for Step Size \( h = 0.1 \)

For a step size of \( h = 0.1 \), the update formula is given by \( y_{n+1} = y_n + 0.1(2x_n - 3y_n + 1) \), which simplifies to \( y_{n+1} = 0.2x_n + 0.7y_n + 0.1 \).

Step 3: Calculating for \( x = 1\) and \( y = 5 \) with \( h = 0.1 \)

Using the step size \( h = 0.1 \), calculate \( y(1.1) \) as: \[ y(1.1) = 0.2(1) + 0.7(5) + 0.1 = 3.8 \] Then, increment \( x \) to \( 1.1 \) and use \( y(1.1) \) calculated to find \( y(1.2) \): \[ y(1.2) = 0.2(1.1) + 0.7(3.8) + 0.1 = 2.98 \].

Step 4: Setting Up for Step Size \( h = 0.05 \)

For a step size of \( h = 0.05 \), the update formula is given by \( y_{n+1} = y_n + 0.05(2x_n - 3y_n + 1) \), which simplifies to \( y_{n+1} = 0.1x_n + 0.85y_n + 0.1 \).

Step 5: Calculating for \( x = 1\) and \( y = 5 \) with \( h = 0.05 \)

Calculate \( y(1.05) \), \( y(1.1) \), \( y(1.15) \), and \( y(1.2) \) using the update formula:- \[ y(1.05) = 0.1(1) + 0.85(5) + 0.1 = 4.4 \] - \[ y(1.1) = 0.1(1.05) + 0.85(4.4) + 0.1 = 3.895 \] - \[ y(1.15) = 0.1(1.1) + 0.85(3.895) + 0.1 = 3.47075 \] - \[ y(1.2) = 0.1(1.15) + 0.85(3.47075) + 0.1 = 3.11514 \]

Key Concepts

Numerical Methods

Numerical methods are powerful mathematical tools used to find approximate solutions to complex problems that are difficult to solve analytically. The purpose of these methods is to simplify calculations and provide practical solutions.

• They are especially useful for problems involving calculus, such as those with difficult integrals or differential equations.
• Since not all mathematical equations can be solved with an exact formula, numerical methods like Euler's Method help estimate the value of unknown variables.
• These methods transform complicated equations into iterative procedures that computers can easily handle.

In applying numerical methods, the goal is to iteratively approach a solution by breaking down the equation into manageable parts.
Each iteration helps to get closer to the accurate value. Euler's Method is one such numerical method that is widely applied to differential equations, providing step-by-step estimations.

Step Size

The step size, often denoted by the letter \( h \), is a crucial parameter in numerical methods like Euler's Method. It determines how quickly or slowly the calculations advance along the interval.

• A smaller step size means more calculations and better approximation, but it requires more computational effort.
• A larger step size is faster and less computationally demanding, but it may reduce the accuracy of the outcome.

In our original exercise, we use two step sizes: \( h = 0.1 \) and \( h = 0.05 \).
The smaller step size offers a finer approximation of the differential equation over a given range. It's like taking smaller, careful steps to reach a destination versus taking larger leaps.
Choosing the right step size is a balancing act between the desired accuracy and the available computational resources.

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. These equations are essential in modeling real-world phenomena where change is continuous.

• They describe various physical and natural systems, such as motion, electricity, and heat transfer.
• The solutions to differential equations represent a way to predict behavior over time.
• While some differential equations can be solved analytically, others require numerical methods.

In this context, Euler's Method, part of our original problem, helps approximate solutions to ordinary differential equations.
By applying Euler's Method, we use an iterative approach to estimate the behavior of complex systems described by these equations.
Such numerical approximations are indispensable in cases where exact solutions are impossible to determine analytically.