Final answer:
The local truncation error in Euler's method is related to the step size and the second derivative of the solution. Without the second derivative, the exact formula cannot be provided, but the general form involves the square of the step size and the second derivative at a point within the current interval.
Step-by-step explanation:
To find the local truncation error in the nth step of Euler's method for the given initial-value problem y' = 2x - 3y + 1, y(1) = 5, we first need to understand what local truncation error represents. It measures the error made within one step of the numerical method, assuming the previous step was exact.
For Euler's method, the local truncation error at step n can be expressed in terms of the function's second derivative, which can be computed from the original differential equation.
Unfortunately, without the second derivative, we cannot provide a specific formula for the error. However, in general terms, the local truncation error for Euler's method is proportional to the square of the step size (h^2) and the second derivative of the actual solution with respect to x, evaluated at the point xn.
The formula for local truncation error in Euler's method is typically error = (h^2/2) * y''(xc), where y''(x) is the second derivative of the actual solution at some point xc within the interval [xn, xn+1], and h is the step size. To find the exact error for this problem, you would need to compute y''(x) using the given analytic solution.