Euler’s method on more general grids Suppose the solution of the initial value problem y'(t)=f(t,y),y(a)=A is to be approximated on the interval [a, b].
b. Write the first step of Euler’s method to compute u1.

Verified step by step guidance

1. Recall that Euler's method is a numerical technique to approximate the solution of the initial value problem \(y'(t) = f(t, y)\) with initial condition \(y(a) = A\) over the interval \([a, b]\).
2. Choose a step size \(h = t_1 - t_0\), where \(t_0 = a\) and \(t_1\) is the next point in the grid where we want to approximate the solution.
3. The first approximation \(u_0\) is given by the initial condition: \(u_0 = A\).
4. Use the Euler's method formula to compute the next value \(u_1\): \[u_1 = u_0 + h \cdot f(t_0, u_0)\]
5. This formula means you take the current value \(u_0\) and add the product of the step size \(h\) and the slope \(f(t_0, u_0)\) to estimate the solution at \(t_1\).

Key Concepts

Initial Value Problem (IVP)

An initial value problem involves finding a function y(t) that satisfies a differential equation y'(t) = f(t, y) along with a given initial condition y(a) = A. The solution is sought over an interval [a, b], starting from the known initial value.

Euler’s Method

Euler’s method is a numerical technique to approximate solutions of differential equations by using tangent line approximations. Starting from the initial point, it estimates the next value by moving along the slope given by f(t, y) multiplied by a step size.

Step Size and Grid Points

The interval [a, b] is divided into discrete points called a grid, where the step size determines the distance between these points. In Euler’s method, the step size affects the accuracy of the approximation, and the first step computes u1 using the initial value and the slope at the initial point.