29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.


b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.


y′(t) = 2t + 1, y(0) = 0; y(t) = t² + t

Verified step by step guidance

1. Identify the given differential equation and initial condition: \(y'(t) = 2t + 1\) with \(y(0) = 0\).
2. Note the exact solution provided: \(y(t) = t^{2} + t\).
3. Recall Euler's method formula for approximating the solution at discrete points: \(y_{n+1} = y_n + h \cdot f(t_n, y_n)\), where \(h\) is the step size and \(f(t, y) = y'(t)\).
4. Choose the step size \(h\) based on the points of interest, here from \(t=0\) to \(t=0.2\) and then to \(t=0.4\), so \(h=0.2\). Use Euler's method iteratively to find approximate values \(y(0.2)\) and \(y(0.4)\).
5. Calculate the errors at \(t=0.2\) and \(t=0.4\) by subtracting the Euler approximations from the exact solution values: \(\text{Error} = |y_{exact}(t) - y_{Euler}(t)|\).

Key Concepts

Euler's Method

Euler's method is a numerical technique to approximate solutions of initial value problems for ordinary differential equations. It uses a stepwise approach, estimating the next value by moving along the slope given by the differential equation at the current point. The accuracy depends on the step size, with smaller steps generally yielding better approximations.

Exact Solution and Error Calculation

The exact solution is the precise function satisfying the differential equation and initial condition. To compute errors in Euler's method, subtract the approximate value from the exact value at specific points. This difference quantifies the accuracy of the numerical approximation.

Initial Value Problems (IVPs)

An initial value problem specifies a differential equation along with a starting value for the unknown function. Solving an IVP involves finding a function that satisfies both the differential equation and the initial condition, providing a unique solution curve.