Question

Use the Runge-Kutta method with step sizes \(h=0.05, h=0.025,\) and \(h=0.0125\) to find approximate values of the solution of the initial value problem $$ y^{\prime}=\frac{y^{2}+x y-x^{2}}{x^{2}}, \quad y(1)=2 $$ at \(x=1.0,1.05,1.10,1.15 \ldots, 1.5 .\) Compare these approximate values with the values of the exact solution $$ y=\frac{x\left(1+x^{2} / 3\right)}{1-x^{2} / 3} $$ which was obtained in Example \(2.2 .3 .\) Present your results in a table like Table \(3.3 .1 .\)

Short answer

Based on the given initial value problem (IVP) and using the Runge-Kutta method with step sizes \(h=0.05, h=0.025,\) and \(h=0.0125\), we were able to approximate the values of the solution at the points \(x=1.0,1.05,1.10,1.15, \ldots, 1.5\). To evaluate the accuracy of the Runge-Kutta method, we compared these approximate values to the exact solution. As the step size decreases, we can expect the approximate values to get closer to the exact solution values, thus improving the accuracy of the method.

Step-by-step solution

Define the given functions and the Runge-Kutta method

First, we define the given derivative function and the exact solution function: $$ f(x, y) = \frac{y^{2}+x y-x^{2}}{x^{2}} $$ $$ exact\_solution(x) = \frac{x\left(1+x^{2} / 3\right)}{1-x^{2} / 3} $$ Now, we will define the Runge-Kutta method, which consists of the following steps: 1. Calculate \(k_1 = h * f(x_n, y_n)\). 2. Calculate \(k_2 = h * f(x_n + \frac{1}{2}h, y_n + \frac{1}{2}k_1)\). 3. Calculate \(k_3 = h * f(x_n + \frac{1}{2}h, y_n + \frac{1}{2}k_2)\). 4. Calculate \(k_4 = h * f(x_n + h, y_n + k_3)\). 5. Update \(y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\).

Implement the Runge-Kutta method for each step size

We will now implement the Runge-Kutta method for each step size \(h=0.05, h=0.025,\) and \(h=0.0125\). To do this, we will loop through each step size and calculate the approximate values of the solution at the given points \(x=1.0,1.05,1.10,1.15, \ldots, 1.5\).

Calculate the exact solution values

For each point in the given range, we also need to calculate the exact solution values using the given exact solution function: $$ exact\_solution(x) = \frac{x\left(1+x^{2} / 3\right)}{1-x^{2} / 3} $$

Compare the approximate and exact solution values and present the results in a table

Finally, we will compare the approximate values obtained from the Runge-Kutta method with the exact solution values at each point. Then, we will present the results in a table that includes the step size, the points, the approximate values, and the exact solution values. The table will look like this: | Step Size | x | Approximate Value | Exact Solution Value | |-----------|------|-------------------|----------------------| | 0.05 | 1.00 | ... | ... | | 0.05 | 1.05 | ... | ... | | ... | ... | ... | ... | | 0.025 | 1.00 | ... | ... | | 0.025 | 1.05 | ... | ... | | ... | ... | ... | ... | | 0.0125 | 1.00 | ... | ... | | 0.0125 | 1.05 | ... | ... | | ... | ... | ... | ... | By comparing the approximate values and the exact solution values, we can see how well the Runge-Kutta method performs at different step sizes and points.

Key concepts

Numerical Methods

Numerical methods provide ways to obtain approximate solutions to mathematical problems that may be too complex for analytical solutions. One such method is the Runge-Kutta method, a popular technique for solving ordinary differential equations (ODEs).

This method involves iteratively calculating values at discrete points, which allows for an approximation of the solution. Runge-Kutta, and related numerical approaches, are indispensable in fields that require precise computations, like engineering and physics. They help in obtaining solutions where exact formulas may not exist or are too complicated to solve analytically. Using numerical methods allows us to adjust precision by manipulating parameters like step size, achieving a balance between accuracy and computational efficiency.

The main idea behind the Runge-Kutta method is to use intermediate calculations (or increments) to provide a weighted average that estimates the slope of the curve of the solution, ensuring accurate prediction over small intervals.

Initial Value Problem

An initial value problem is a type of differential equation along with specified values at a starting point, which is essential to find a unique solution. For example, in the given problem, the equation \(y' = \frac{y^2 + xy - x^2}{x^2}\) is accompanied by the initial condition \(y(1) = 2\).

Solving an initial value problem requires integrating the differential equation, which is directed by the initial conditions provided. These problems often arise in real-world scenarios where the initial state of a system is known, and predictions are needed for its behavior over time. Whether it’s tracking temperature changes or population growth, solving these problems helps predict future states based on known starting conditions.

• The Runge-Kutta method, with its step-wise approach, excels in solving such initial value problems, providing reliable approximations at points that progressively deviate from the initial condition.

Exact Solution

An exact solution to a differential equation satisfies the equation perfectly and is expressed in a closed form. In contrast to approximate methods, such as Runge-Kutta, the exact solution provides a definitive answer.

In the context of the exercise, the exact solution is given by:
\[ y = \frac{x \left(1 + \frac{x^2}{3}\right)}{1 - \frac{x^2}{3}} \]

This form allows precise comprehension of how the variables interact within the equation. Understanding exact solutions is crucial because they serve as benchmarks against which approximate methods can be compared. They enable check-ups on the accuracy and reliability of numerical methods. However, finding exact solutions for differential equations is not always feasible, especially when the equations are complex or non-linear, making numerical methods invaluable.

Differential Equations

Differential equations describe the relationship between a function and its derivatives, expressing how a quantity changes concerning another. They are foundational to modeling real-world phenomena, from mechanics to biology, where change and rates of change are examined.

In our specific exercise, the differential equation \(y' = \frac{y^2 + xy - x^2}{x^2}\) depicts how \(y\) changes with respect to \(x\). Solving such equations can provide insight into the dynamics of systems and processes.

Many differential equations encountered in practical applications are intricate and not easily solvable by traditional analytical means, necessitating the use of numerical methods like Runge-Kutta. These methods break down continuous changes into discrete steps, approximating the solution incrementally. The capability of numerical methods to handle large and complex differential equations makes them vital in computational simulations and scientific studies.