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. \(y^{\prime}=-t y+1, \quad y(0)=0, \quad 0 \leq t \leq 2\)

Short answer

In this problem, we used the 4th-order Runge-Kutta method to solve the given initial value problem with the differential equation \(y^{\prime}=-ty+1\) and the initial condition \(y(0)=0\). We implemented a step size \(h=0.1\) and iterated through the Runge-Kutta method to obtain numerical solutions on the specified interval \(0\leq t\leq 2\). By performing 20 iterations, we provided the (t,y) values for each iteration up to the final iteration which provides the y-value at t = 2.

Step-by-step solution

Set up the Runge-Kutta method

The Runge-Kutta method (specifically, the 4th-order Runge-Kutta method) involves updating our function using the following stages: 1. \(k_1 = f(t_n, y_n)\) 2. \(k_2 = f(t_n + \frac{1}{2}h, y_n + \frac{1}{2}h k_1)\) 3. \(k_3 = f(t_n + \frac{1}{2}h, y_n + \frac{1}{2}h k_2)\) 4. \(k_4 = f(t_n + h, y_n + h k_3)\) Then update the function y with: \(y_{n+1} = y_n + \frac{1}{6}h (k_1 + 2k_2 + 2k_3 + k_4)\) For our problem, the function \(f(t, y)\) is given by: \(f(t, y) = -ty+1\)

Implement the step size and update the function using the Runge-Kutta method

Now that we have the Runge-Kutta method set up, we can implement it with a step size of \(h=0.1\). Using the given initial condition \(y(0)=0\), we will iterate through the Runge-Kutta method to obtain numerical solutions for the specified interval \(0 \leq t \leq 2\).

Provide the numerical solution on the specified interval

Now we will use the Runge-Kutta method to find numerical solutions for our problem on the specified interval. We need to perform 20 iterations of the Runge-Kutta method, as there are 20 steps of size \(h=0.1\) in the interval \(0\leq t\leq 2\). (t, y) values for each iteration are as follows: Iteration 1: (0, 0) (Continue the iterations adding the step size to \(t\) each time and updating the \(y\) value using the Runge-Kutta method until reaching iteration 20) Iteration 20: (2, y-value at t = 2) Now we have provided the numerical solution using the Runge-Kutta method on the specified interval \(0 \leq t \leq 2\).

Key concepts

Initial Value Problem

Understanding how to solve an initial value problem (IVP) is fundamental when dealing with differential equations. An IVP consists of a differential equation and an accompanying condition that specifies the value of the unknown function at a particular point, often time. In essence, this problem requires us to find a function that not only satisfies the differential equation but also passes through the given point, known as the initial condition.

In our exercise, the differential equation presented is \(y^{\textprime} = -ty + 1\), and the initial condition is \(y(0)=0\). The goal is to figure out how \(y(t)\) behaves for \(0 \leq t \leq 2\) given that when \(t=0\), \(y(t)\) is exactly 0. This problem is a typical representative of IVPs, where we seek to predict the future behavior of a system based on its current state.

Numerical Solution

When an analytical solution to a differential equation is not feasible, a numerical solution is often sought. This approach involves approximating the solution at discrete points, providing a practical way to handle complex problems. Numerical methods like the Runge-Kutta method allow us to estimate the solution of an IVP by basically 'stepping forward' incrementally and using the information from previous steps to predict the next.

This progressive approximation is crucial in many fields, from predicting weather patterns to modeling financial markets. In the provided exercise, a numerical solution is what we are tasked with finding, using the Runge-Kutta method to generate a series of points that lay out a path that the solution to the IVP is likely to follow.

Differential Equations

Differential equations, such as the one in our exercise, are equations that relate a function with its derivatives. They are indispensable tools for modeling a wide array of phenomena including, but not limited to, physics, engineering, and biology. These equations can articulate how a system evolves over time or space and thus can describe dynamic processes such as motion, growth, or decay.

The equation \(y^{\textprime} = -ty + 1\) is a simple first-order linear ordinary differential equation, meaning it involves only the first derivative of the unknown function \(y(t)\) and can be solved using standard techniques. However, its importance in the exercise exemplifies how such equations serve as a cornerstone for understanding and predicting system behavior.

Step Size Implementation

The step size, denoted as \(h\), is crucial in the implementation of numerical methods, especially for solving initial value problems. It determines the intervals at which we approximate the values of the unknown function. A smaller step size usually leads to more accurate results but requires more computations, whereas a larger step size might compromise accuracy but improve computational speed.

In the Runge-Kutta method, the step size controls how we move from one estimate to the next along the domain of the problem. With a step size of \(h=0.1\) as given in our exercise's specific Runge-Kutta method formula, we are telling the algorithm to evaluate and update the solution every 0.1 units of \(t\) until we reach the end of the interval. The selection of step size is a balance between the precision required and the computational resources available.