Question

Consider the initial value problem \(y^{\prime}(t)=t^{2}-3 y^{2}, y(3)=1\) What is the approximation to \(y(3.1)\) given by Euler's method with a time step of \(\Delta t=0.1 ?\)

Short answer

Question: Using Euler's method with a time step of 0.1, find the approximation of y(3.1) for the initial value problem \(y^{\prime}(t)=t^{2}-3y^{2}\), with \(y(3)=1\). Answer: The approximation of y(3.1) using Euler's method with a time step of 0.1 is 1.6.

Step-by-step solution

Identify the differential equation and initial values

We are given the following initial value problem: $$ y^{\prime}(t)=t^{2}-3y^{2},\quad y(3)=1 $$

Initialize variables for Euler's method

The given time step is \(\Delta t = 0.1\). The initial point is \((t_0, y_0) = (3, 1)\).

Apply Euler's method

Define \(t_{n+1} = t_n + \Delta t\) and \(y_{n+1} = y_n + y^{\prime}(t_n) \Delta t\). Now, we will use Euler's method with one step of time: 1. At \(n=0\): \(t_0 = 3\), \(y_0 = 1\), \(\Delta t = 0.1\) Now, we have \(y^{\prime}(t_0) = t_0^2 - 3y_0^2 = 3^2 - 3(1^2) = 6\). Therefore, \(y_1 = y_0 + y^{\prime}(t_0) \Delta t = 1 + 6(0.1) = 1.6\) and \(t_1 = t_0 + \Delta t = 3 + 0.1 = 3.1\).

Find the approximation of the function at t = 3.1

We have found that: $$ y(3.1) \approx y_1 = 1.6 $$ Thus, the approximation of y(3.1) using Euler's method with a time step of 0.1 is 1.6.

Key concepts

Initial Value Problem

An initial value problem in mathematics involves solving a differential equation with a given initial value. Specifically, it means determining a function that satisfies the differential equation, along with initial conditions necessary for a unique solution.
For example, consider when we are given a differential equation, such as \( y'(t) = t^2 - 3y^2 \), along with an initial condition like \( y(3) = 1 \). This pair collectively forms what's called an initial value problem. The challenge here is to find the function \( y(t) \) that meets both the differential equation and the initial condition.
Initial value problems are foundational in mathematics, allowing us to model countless real-world phenomena from physics, biology, economics, and beyond. They offer a snapshot starting point from which behavior over time or intervals is calculated and understood.

Differential Equation

A differential equation is a mathematical statement involving an unknown function and its derivatives. These equations express relationships between functions and their rates of change, capturing the essence of dynamic systems in the process.
For instance, differential equations can describe the motion of a car, the growth of a population, or the changes in investment returns over time. In this exercise, the differential equation presented is \( y'(t) = t^2 - 3y^2 \). It indicates how the rate of change of \( y \) over time \( t \) is a function of both \( t \) itself and the square of \( y \).
Solving these equations either analytically (when solutions can be expressed in exact formulas) or numerically (using approximation methods) is key to understanding complex dynamic relationships. Differential equations play a vital role in fields like engineering, physics, and applied sciences, serving as core guides to solve real-world challenges.

Numerical Approximation

Numerical approximation is a powerful tool used when exact solutions are either impossible or impractical to obtain. It refers to the use of computational algorithms to estimate the values of unknowns such as functions, solutions, or loads in a system.
Euler's Method is one prevalent technique of numerical approximation, especially useful for solving initial value problems. It approximates the solution of a differential equation using a step-by-step approach. From an initial point, each step uses the slope of the equation to find the next value.
Here, we utilized Euler's Method with a time step, \( \Delta t = 0.1 \), to approximate \( y(3.1) \) beginning from the initial condition \( y(3) = 1 \). By calculating the slope and moving in small time intervals, the method advances incrementally to a new estimated value, \( y(3.1) \approx 1.6 \). This approach is essential in scenarios where analytical solutions are challenging and in contexts requiring simulation or predictive modeling.