33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
a. Approximate the value of y(T) using Euler’s method with the given time step on the interval [0,T].


y′(t) = t/y, y(0) = 4; Δt = 0.1, T = 2; y(t) = √(t² + 16)

Verified step by step guidance

1. Identify the differential equation and initial condition: \(y'(t) = \frac{t}{y}\) with \(y(0) = 4\).
2. Set the step size \(\Delta t = 0.1\) and the interval from \(t=0\) to \(T=2\). Determine the number of steps \(n = \frac{T - 0}{\Delta t} = \frac{2}{0.1} = 20\).
3. Use Euler's method formula to approximate \(y\) at each step: \(y_{k+1} = y_k + \Delta t \cdot f(t_k, y_k)\), where \(f(t, y) = \frac{t}{y}\).
4. Start with the initial values \(t_0 = 0\) and \(y_0 = 4\). For each step \(k\) from 0 to 19, compute \(y_{k+1}\) using the formula and update \(t_{k+1} = t_k + \Delta t\).
5. After completing all steps, the value \(y_{20}\) will be the Euler approximation of \(y(2)\). Compare this approximation to the exact solution \(y(t) = \sqrt{t^2 + 16}\) if desired.

Key Concepts

Euler's Method

Euler's method is a numerical technique to approximate solutions of first-order differential equations. It uses a stepwise approach, updating the solution by moving along the slope given by the derivative at each step. This method is especially useful when an exact solution is difficult to find.

Initial Value Problems (IVP)

An initial value problem specifies the value of the solution at a starting point, allowing the differential equation to be solved uniquely. Here, y(0) = 4 sets the initial condition, which is essential for applying Euler's method to approximate y(t) over the interval.

Differential Equation and Exact Solution

The differential equation y′(t) = t/y relates the rate of change of y to t and y itself. The exact solution y(t) = √(t² + 16) provides a benchmark to compare the accuracy of Euler's approximation, helping to understand the method's precision and error.