Question

You may want to save the results of these exercises, sincewe'll revisit in the next two sections.Use Euler's method to find approximate values of the solution of the given initial value problem at the points $x_i=x_0+ih,$ where $x_0$ is the point wher the initial condition is imposed and $i=1,2,3$. The purpose of these exercises is to familiarize you with the computational procedure of Euler's method. $$y^{\prime }=2x^2+3y^2-2,y(2)=1;h=0.05$$

Short answer

The approximate values for the given initial value problem are $y_1\approx 1.5$, $y_2\approx 2.2745$, and $y_3\approx 3.399025$.

Step-by-step solution

Write down the given differential equation, initial condition, and step size.

The given initial value problem is: $$y^{\prime }=2x^2+3y^2-2,y(2)=1$$ The step size used for Euler's method is $h=0.05$.

Use Euler's method formula to find $y_{i+1}$ given $y_i$ and $x_i$

For any point $(x_i,y_i)$, Euler's method can be used to approximate the solution at the next point $y_{i+1}$ using the formula: $$y_{i+1}=y_i+hf(x_i,y_i)$$ where $h$ is the step size and $f(x_i,y_i)$ refers to the value of the derivative function at the point $(x_i,y_i)$. In this case: $$f(x_i,y_i)=2x_i^2+3y_i^2-2$$

Compute $y_1$ using the initial condition $y(2)=1$ and $h=0.05$

First, let's calculate $x_1=x_0+h=2+0.05=2.05$. Then compute $f(x_0,y_0)=2(2{)}^2+3(1{)}^2-2=10$. Using Euler's method formula: $$y_1=y_0+h{i}_1\ast f(x_0,y_0)=1+0.05\ast 10=1.5$$

Compute $y_2$ using $y_1=1.5$ and $h=0.05$

First, let's calculate $x_2=x_0+2h=2+2\ast 0.05=2.1$. Then compute $f(x_1,y_1)=2(2.05{)}^2+3(1.5{)}^2-2=15.49$. Using Euler's method formula: $$y_2=y_1+h{i}_2\ast f(x_1,y_1)=1.5+0.05\ast 15.49=2.2745$$

Compute $y_3$ using $y_2=2.2745$ and $h=0.05$

First, let's calculate $x_3=x_0+3h=2+3\ast 0.05=2.15$. Then compute $f(x_2,y_2)=2(2.1{)}^2+3(2.2745{)}^2-2=22.4805$. Using Euler's method formula: $$y_3=y_2+h{i}_3\ast f(x_2,y_2)=2.2745+0.05\ast 22.4805=3.399025$$ So, using Euler's method with a step size of $h=0.05$, the approximate values for the given initial value problem are $y_1\approx 1.5$, $y_2\approx 2.2745$, and $y_3\approx 3.399025$.

Key concepts

Numerical Methods

Numerical methods are techniques used to find approximate solutions to mathematical problems when exact solutions are difficult or impossible to obtain through analytical means. These methods are often employed when dealing with complex equations that are challenging to solve systematically.
Euler's method, specifically, is a first-order numerical procedure used for solving ordinary differential equations (ODEs) with a given initial value. As a simple and straightforward technique, it allows us to approximate the value of functions step by step.

• It requires the differential equation and an initial condition to begin.
• The solution is constructed incrementally, creating a series of points that approximate the curve of the function.
• The step size, denoted by $h$, plays a critical role in determining the accuracy of the approximation. Smaller steps usually result in more accurate solutions, though they require more calculations.

Understanding numerical methods like Euler’s method is crucial for tackling real-world problems where exact solutions are not feasible. Its simplicity makes it a useful tool in introductory courses covering differential equations and computational mathematics.

Initial Value Problem

An initial value problem (IVP) in mathematics involves solving a differential equation along with a specified value, or condition, at a starting point. This initial condition acts as a starting reference, guiding the solution in a defined direction.
In the problem we discussed, the initial condition is $y(2)=1$, which means at $x=2$, the value of $y$ is 1. This serves as the seed for our solution, allowing methods like Euler's to propagate the calculated values forward.

• An IVP comprises a differential equation and an initial condition.
• These problems are foundational in numerous academic and applied fields such as physics, engineering, and finance, since many models use differential equations with known conditions at certain points.
• Numerical methods, including Euler's method, are ideal for solving IVPs when exact analytical solutions are unachievable.

Working through initial value problems gives students a solid understanding of how dynamic systems evolve over time, based on their initial states.

Differential Equations

Differential equations form the backbone of modeling how physical systems change over time or space. These equations relate functions to their derivatives, indicating how a function's value evolves as the independent variable, like time or space, changes.
The equation given is $y^{\prime }=2x^2+3y^2-2$. It is a first-order differential equation, meaning it involves the first derivative of $y$ with respect to $x$.

• Differential equations can be complex, and exact solutions are not always easy to determine.
• They come in several types: ordinary differential equations (ODEs), partial differential equations (PDEs), linear and non-linear, based on their characteristics.
• Understanding how to interpret and set up such equations is crucial for fields where modeling change is essential, like physics or biology.

When combined with methods like Euler's, differential equations become powerful tools for prediction and analysis, allowing scholars and practitioners to simulate real-world phenomena accurately.