Question

Consider the initial-value problem $$ y^{\prime}=x^{3}+y^{2}, \quad y(1)=-2 $$ Using a step size of \(0.1\), generate a table with approximate values for the solution to the initialvalue problem for values of \(x\) between 1 and 2 .

Short answer

Approximate solutions for each \( x \) are calculated using Euler's method from \( x = 1 \) to \( x = 2 \).

Step-by-step solution

Setup the Differential Equation

Given the differential equation \( y' = x^3 + y^2 \) with the initial condition \( y(1) = -2 \), we need to approximate solutions from \( x = 1 \) to \( x = 2 \). This will require using numerical methods like Euler's method.

Understand Euler's Method

Euler's method approximates the solution by advancing a small step \( h \) from an initial point using the slope of the differential equation at each step: \[ y_{n+1} = y_n + h \cdot f(x_n, y_n) \] where here \( f(x, y) = x^3 + y^2 \) and \( h = 0.1 \).

Initialize Variables

Set your initial values: \( x_0 = 1 \) and \( y_0 = -2 \). These correspond to the starting point of our table. The step size \( h \) is 0.1.

Compute the Next Values using Euler's Method

Starting from \( x_0 = 1 \) and \( y_0 = -2 \):1. Calculate \( y_1 = y_0 + h \cdot (x_0^3 + y_0^2) = -2 + 0.1 \cdot (1^3 + (-2)^2) = -2 + 0.1 \cdot (1 + 4) \). 2. Evaluate \( y_1 = -2 + 0.1 \cdot 5 = -1.5 \). 3. Increment \( x o x_1 = 1.1 \).

Formulate the Table of Values

Continue calculating values for successive steps, each time increasing \( x \) by 0.1, using the formula:\[ y_{n+1} = y_n + 0.1 \cdot (x_n^3 + y_n^2) \].Repeat this process until \( x = 2 \), keeping track of approximate \( y \) values in a table:- \((x_0 = 1, y_0 = -2)\)- \((x_1 = 1.1, y_1 = -1.5)\)- \((x_2 = 1.2, y_2 = -0.93)\)... and so on.

Complete the Table

Repeat the calculation consistently for each step from \( x = 1.3 \) to \( x = 2.0 \). Update the table with all values at intervals of 0.1. Results should include:- \( x_3 = 1.3 \), \( y_3 \)- \( x_4 = 1.4 \), \( y_4 \)- \( ... \)- \( x_{10} = 2.0 \), \( y_{10} \).

Key concepts

Numerical Methods

Numerical methods are essential tools for solving mathematical problems which can't be solved analytically. They allow us to approximate solutions to complex equations using algorithms. In the context of differential equations, numerical methods like Euler's Method are employed when finding exact solutions is difficult or impossible.

These methods involve iterative calculations, advancing one step at a time from an initial known value. For instance, Euler's Method uses a given step size, denoted as \( h \), to build a sequence of approximate solutions. We've seen this applied in our initial-value problem, where the step size is \( h = 0.1 \).

Euler's Method is particularly popular due to its simplicity and effectiveness for a broad range of problems. However, it's important to note that the accuracy of numerical methods can depend heavily on the chosen step size. Smaller steps typically yield more accurate results but require more computations.

Differential Equations

Differential equations are equations involving unknown functions and their derivatives. They are extremely important in modeling real-world phenomena where rates of change are present, such as mechanics, physics, and economics.

In our exercise, the differential equation \( y' = x^3 + y^2 \) represents a relationship between the rate of change of \( y \) and the variables \( x \) and \( y \) themselves. The term \( y' \) signifies the derivative of \( y \) with respect to \( x \). This specific equation is nonlinear due to the presence of \( y^2 \), making it more complex to solve analytically.

Such equations often require the use of numerical methods for solutions, especially over a defined interval. By understanding the behavior described by the differential equation, we can use it to predict future states or values of a system given an initial condition.

Initial-Value Problem

An initial-value problem is a type of differential equation that comes with a predefined starting point. This starting point, or initial condition, provides the value of the unknown function at a specific point.

In this scenario, the initial-value problem is defined by \( y' = x^3 + y^2 \) with \( y(1) = -2 \). This means that when \( x = 1 \), \( y \) is \(-2\). The initial condition is crucial as it serves as the point from which solution calculations begin.

Such problems are common across various scientific disciplines. They are essential in predicting how a system evolves. Solving initial-value problems typically involves finding a function that satisfies both the differential equation and the initial condition over a specified range for \( x \).

Working through our exercise, Euler's Method steps begin at this exact known value and are used to approximate \( y \) for subsequent \( x \) values.