91Ó°ÊÓ

Ordinary differential equations (ODEs) are equations that involve a function and its derivatives. They describe how a particular quantity changes over time and are fundamental in various fields like physics, engineering, and biology. For example, if we know how the rate of change of a population depends on the current population size, we can use an ODE to model this. This particular exercise involves an ODE given by \({y}' = t^2 y\) with an initial condition of \(y(0) = -2\).The solution to an ODE helps us understand the behavior of the system being modeled. However, many ODEs don't have simple analytical solutions. That's where numerical methods like Euler's method come in, allowing us to approximate solutions even when exact solutions are impossible to find.

Numerical approximation methods, such as Euler's method, provide ways to approximate the solutions of problems that may be difficult or impossible to solve analytically. Euler's method specifically is used to approximate the solutions to ordinary differential equations by using the idea of taking small steps along the curve defined by the ODE.
To approximate the solution, Euler's method follows these steps:

• Initialization: Start from a known initial value.
• Iteration: Update the value using the ODE's rate of change. For each step from one point to the next, apply the formula:
  \(y_{n+1} = y_n + h f(t_n, y_n)\)
  where \(h\) is the step size, \(y_n\) is the current value, and \(f(t_n, y_n)\) is the function forming the ODE's right-hand side.
• Repetition: Repeat the iteration for the desired number of steps.

In the given exercise, with \(n=2\) steps in the interval \(0 \leq t \leq 1\), the step size \(h = \frac{1-0}{2} = 0.5\). This initializes the starting value \(t_0 = 0\), \(y_0 = -2\) and computes subsequent values using the iteration formula.

An initial value problem (IVP) is a type of differential equation together with a specified value, called the initial condition, at a given point. The solution to an IVP provides a function that satisfies the differential equation and passes through the initial condition.
In this exercise, the initial value problem is defined by \( y' = t^2 y \) along with the initial condition \( y(0) = -2 \). The goal of solving an IVP is to find the function \( y(t) \) for all \( t \geq 0 \) that satisfies both the differential equation and the initial condition.
Here is a breakdown of how Euler's method is applied to solve this initial value problem:

• We start at \( t_0 = 0 \) with \( y_0 = -2 \).
• Using the step size \( h = 0.5 \), we proceed to calculate \( y_1 \) and \( y_2 \) iteratively.
• First step: \( y_1 = y_0 + h f(t_0, y_0) = -2 + 0.5 \times (0^2 \times -2) = -2 \).
• Second step: \( y_2 = y_1 + h f(t_1, y_1) = -2 + 0.5 \times (0.5^2 \times -2) = -2.25 \).

Therefore, Euler's method approximates \( f(1) = -2.25 \).