Question

Exercises 23-27: A computer or programmable calculator is needed for these exercises. For the given initial value problem, use the Runge-Kutta method \((9)\) with a step size of \(h=0.1\) to obtain a numerical solution on the specified interval. \(y^{\prime}+2 t y=\sin t, \quad y(0)=0, \quad 0 \leq t \leq 3\)

Short answer

Based on this step-by-step solution, write a short answer: To solve the given initial value problem using the RK4 method with a step size of 0.1, first define the differential equation and initial condition. Then, set up the RK4 formulas and iterate the method by applying the formulas step by step throughout the specified interval (0 ≤ t ≤ 3). Finally, obtain the numerical solution to the initial value problem.

Step-by-step solution

Define the differential equation and initial condition

First, define the given differential equation and initial condition: $$ y^{\prime}(t) = -2ty(t) + \sin t \quad \text{and} \quad y(0) = 0 $$ We want to solve this for the interval \(0 \le t \le 3\) with a step size \(h = 0.1\).

Set up the Runge-Kutta method's formulas

The RK4 method involves the following formulas: $$ \begin{aligned} k_1 &= h \cdot f(t_n, y_n)\\ k_2 &= h \cdot f(t_n + \frac{h}{2}, y_n + \frac{k_1}{2})\\ k_3 &= h \cdot f(t_n + \frac{h}{2}, y_n + \frac{k_2}{2})\\ k_4 &= h \cdot f(t_n + h, y_n + k_3)\\ y_{n+1} &= y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4) \end{aligned} $$ where \(f(t, y) = -2ty + \sin t\) in our case.

Iterate the RK4 method by applying the formulas step by step

Now, apply the RK4 method to iterate through the given steps and interval: 1. Start with the initial values \(t_0 = 0\) and \(y_0 = 0\). Calculate \(k_1, k_2, k_3, k_4\) and then find the new value \(y_1\). 2. Continue to iterate \(y_{n+1}\) by using the RK4 formulas until you reach the end of the interval, i.e., \(t = 3\).

Obtain the numerical solution

After iteratively calculating each \(t_n\) and \(y_n\) in the specified interval, you will obtain a numerical solution to the given IVP. In summary, the steps to solve the given IVP \(y^{\prime} + 2 t y = \sin t, \quad y(0)=0, \quad 0 \leq t \leq 3\) using the RK4 method with a step size of \(h = 0.1\) are: 1. Define the differential equation and initial condition. 2. Set up the Runge-Kutta method's formulas 3. Iterate the RK4 method by applying the formulas step by step. 4. Obtain the numerical solution.

Key concepts

Numerical Analysis

Numerical analysis is the branch of mathematics that focuses on algorithms for solving problems that involve continuous variables. Such problems often require the use of computers since analytical solutions are not always possible. Numerical methods approximate solutions to an acceptable degree of accuracy, given limitations of computational power and time.

In the context of differential equations, numerical analysis provides techniques such as the Runge-Kutta method, which is used for approximating solutions to initial value problems (IVPs). The Runge-Kutta methods, including the commonly used fourth-order method (RK4), play a pivotal role in enabling us to predict the behavior of systems governed by differential equations when exact solutions cannot be derived algebraically.

The ability to approximate solutions with high accuracy makes numerical analysis an indispensable tool in scientific computing, engineering, economic modeling, and various other fields where differential equations arise.

Initial Value Problem

An initial value problem (IVP) in the context of differential equations is a problem where the solution is determined uniquely by the values of the dependent variable at a specified point—often the starting point of an interval. In simpler terms, an initial value problem asks you to find a function that not only satisfies a given differential equation but also passes through a given point, known as the initial condition.

Solving IVPs is crucial in many applications of mathematics and science because it corresponds to predicting the future state of a system given its current state. For example, the motion of a planet, the growth of a population, or the decay of a radioactive material can be modeled as IVPs. The Runge-Kutta method's strength lies in efficiently and accurately solving IVPs to give us insights into how these physical systems evolve over time. It does this by iteratively computing the values of a function at discrete intervals, constructing a solution that approximates the actual continuous phenomenon.

Differential Equations

Differential equations are mathematical equations that involve the derivatives of a function. They are a central tool in describing the way many natural and engineered systems change and evolve over time. In essence, they represent the mathematical embodiment of the laws of nature, such as those of motion, energy, and force.

However, not all differential equations can be solved explicitly. That's where numerical methods like the Runge-Kutta method become essential. This method transforms the problem of solving a differential equation into a manageable series of steps that approximate the behavior of the equation over an interval. By understanding the fundamental principles behind differential equations, students can better grasp the importance of numerical methods that enable practical solutions and predictions in complex real-world scenarios. The Runge-Kutta method is therefore not just a computational technique but a bridge between the abstract theory of differential equations and the concrete analysis of phenomena that those equations model.