Question

Use the Runge-Kutta method with step sizes \(h=0.1, h=0.05,\) and \(h=0.025\) to find approximate values of the solution of the initial value problem $$ \left(3 y^{2}+4 y\right) y^{\prime}+2 x+\cos x=0, \quad y(0)=1(\text { Exercise } 2.2 .13) $$ at \(x=0,0.1,0.2,0.3, \ldots, 1.0\).

Short answer

The differential equation in the form \(\frac{dy}{dx} = g(x, y)\) is \(y^{\prime}=\frac{-2x-\cos(x)}{3 y^{2}+4 y}\).

Step-by-step solution

Define the function g(x, y)

Given the problem, we have $$ g(x, y) = \frac{-2x-\cos(x)}{3 y^{2}+4 y}. $$

Implement the Runge-Kutta method

Now, let's implement Runge-Kutta method for the 3 different step sizes: \(h=0.1,\, 0.05,\, 0.025\). For each interval, we will iterate through the interval applying the method. For each step size, the method involves calculating the following values: $$ k_1 = hg(x_n, y_n) \\ k_2 = hg(x_n + \frac{h}{2}, y_n + \frac{k_1}{2}) \\ k_3 = hg(x_n + \frac{h}{2}, y_n + \frac{k_2}{2}) \\ k_4 = hg(x_n + h, y_n + k_3) \\ y_{n+1} = y_n + \frac{k_1 + 2k_2 + 2k_3 + k_4}{6} $$ Repeat the process for each step size until we cover all the interval \(x=0, 0.1, 0.2, ... , 1.0\)

Calculate the approximate values

After applying the Runge-Kutta method for each step size and all the interval, we will obtain approximate values of the solution at each requested point, for each step size \(h\).

Key concepts

Initial Value Problem

In the realm of differential equations, an initial value problem is one where the solution to a differential equation is required to satisfy certain conditions at an initial point. This means that the problem comes with a set condition, usually expressed as a specific value that the solution must take at a starting point.
For example, in this exercise, the initial value problem specifies that at the point \(x=0\), the value of \(y\) must be \(1\). Initial value problems are prevalent in fields such as physics and engineering, as they model systems with known states at a specific point in time. These problems often involve calculating further values based on this initial condition, which the Runge-Kutta method helps to approximate.
By setting an initial condition, it becomes possible to ensure a unique solution to the differential equation.

Numerical Methods

Numerical methods are techniques used to approximate mathematical computations. Especially when dealing with complex differential equations, exact solutions are not always feasible or practical. This is where numerical methods play a crucial role.
They involve iterative processes which yield approximate solutions that can be tailored for specific accuracy requirements. The Runge-Kutta method is an example of a numerical method. It is particularly significant in solving initial value problems related to differential equations. By using various step sizes, like those indicated in our problem \(h=0.1, 0.05,\) and \(0.025\), we can achieve more precise approximations of the solution.
The iterative nature of numerical methods ensures that we can adjust and repeat calculations until the desired level of accuracy is achieved, making them indispensable tools in fields such as computational physics and engineering.

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They capture the relationship between varying quantities and their rates of change, making them very powerful in modeling dynamic systems. They are used extensively to describe real-world phenomena, from the motion of planets to population dynamics.In our given problem, the differential equation is expressed as \[ (3 y^{2}+4 y) y^{\prime}+2 x+\cos x=0, \quad y(0)=1 \]This represents a complex relationship between \(x\) and \(y\) as well as their rates of change. Solving this equation provides insights into how \(y\) evolves with \(x\), given the initial condition. The Runge-Kutta method is employed to find approximate solutions to such equations, especially when they cannot be solved analytically. By understanding differential equations and using appropriate methods like Runge-Kutta, students and professionals can predict behaviors in systems accurately.