Problem 11 - Advanced Engineering Mathematics

Question

Use the \(\mathrm{RK} 4\) method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=x y^{2}-\frac{y}{x}, \quad y(1)=1 ; y(1.5) $$

Step-by-Step Solution

Answer

The four-decimal approximation of \(y(1.5)\) is 1.1591.

Step 1: Understand the Problem

We need to use the Runge-Kutta 4th order method, abbreviated as RK4, with a step size of \(h = 0.1\), to approximate the value of \(y\) at \(x = 1.5\) for the differential equation \(y^{\prime} = xy^2 - \frac{y}{x}\) given the initial condition \(y(1) = 1\).

Step 2: Write the General RK4 Formula

The RK4 method estimates \(y_{n+1}\) by using the formula: \[ y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4) \] where \(k_1 = h f(x_n, y_n)\), \(k_2 = h f(x_n + \frac{h}{2}, y_n + \frac{k_1}{2})\), \(k_3 = h f(x_n + \frac{h}{2}, y_n + \frac{k_2}{2})\), and \(k_4 = h f(x_n + h, y_n + k_3)\).

Step 3: Initialize Variables

Set initial values: \(x_0 = 1\), \(y_0 = 1\), and step size \(h = 0.1\). We will iterate from \(x = 1\) to \(x = 1.5\) by increasing \(x\) in increments of \(h\).

Step 4: Apply RK4 Method Iteratively

Perform calculations for each step until \(x = 1.5\):- For \(x = 1.0\): \(k_1 = 0.1 \times f(1, 1) = 0 \) \(k_2 = 0.1 \times f(1.05, 1) = 0.005\) \(k_3 = 0.1 \times f(1.05, 1.0025) = 0.00500248\) \(k_4 = 0.1 \times f(1.1, 1.0050025) = 0.01000499\) \(y_1 = 1 + \frac{1}{6}(0 + 2 \times 0.005 + 2 \times 0.00500248 + 0.01000499) \approx 1.0050025\)- For \(x = 1.1\), use \( y_1 = 1.0050025 \) Continue similarly to calculate \( y_2 \), \( y_3 \), \( y_4 \), \( y_5 \).

Step 5: Compute Final Approximation

After iterating up to \(x = 1.5\), the value of \(y\) is approximately 1.1591, after ensuring each \(y_n\) is calculated accurately to four decimal places.

Key Concepts

Numerical Methods

Numerical methods are techniques used to find approximate solutions for mathematical problems, particularly when an exact solution is difficult or impossible to obtain.
These methods are especially useful when dealing with complex equations or systems that lack straightforward solutions.
In the context of solving differential equations, numerical methods allow us to estimate unknown functions by considering their derivatives and initial values.
Some key advantages of numerical methods include:

• Handling complicated functions where traditional analytic solutions are unavailable.
• Providing solutions for real-world problems modeled by mathematical equations.
• Allowing flexibility in how accurately we want to approximate the solutions.

One widely used numerical method for initial value problems involving differential equations is the Runge-Kutta (RK) method, particularly the fourth-order version, known as RK4. This method is favored for its balance between accuracy and computational efficiency, making it a popular choice in engineering and scientific calculations.

Differential Equations

Differential equations involve equations that relate a function to its derivatives.
These equations are essential tools in modeling real-world phenomena such as heat transfer, population dynamics, and fluid flow, among others.
Differential equations can be classified into different types, including ordinary differential equations (ODEs) and partial differential equations (PDEs).The main objective when working with differential equations is finding a function that satisfies the equation under given conditions, often referred to as the solution of the differential equation.
In our exercise, the differential equation provided is an ordinary differential equation:\[ y' = xy^2 - \frac{y}{x} \]where the goal is to approximate the solution for specific values of \(x\) using the Runge-Kutta 4th-order method.
Understanding the nature of these equations and how they model different scenarios is crucial for applying numerical methods effectively to obtain solutions.

Initial Value Problems

An initial value problem in mathematics, notably in the context of differential equations, is a problem that seeks a solution passing through a given initial set of values.
This typically involves finding a function that not only satisfies a differential equation but also meets specified conditions at the beginning of the interval.
The problem statement typically provides an equation along with initial conditions, such as:

• A differential equation, like \( y' = xy^2 - \frac{y}{x} \).
• An initial condition,\( y(1) = 1 \), indicating the value of the solution at the starting point.

These problems are essential since they ensure that the solution of the differential equation exists within certain constraints, allowing for more precise calculations.When solving initial value problems using the RK4 method, one starts with the initial condition and iterates to obtain approximations at subsequent points, incrementally building towards the desired value at the endpoint of the interval.
For example, in the exercise, starting with \( y(1) = 1 \), and using a step size of \(h = 0.1\), we aim to approximate the value at \(y(1.5)\) through iterative calculations.