Use Euler's Method with increments of $\Delta x=0.1 to approximate the value of y when x=1.3. frac{d y}{d x}=x-1 and y=2 when x=1

Step 1 ：Given the differential equation \(\frac{d y}{d x}=x-1\) and the initial condition y=2 when x=1, we are asked to approximate the value of y when x=1.3 using Euler's method with increments of \(\Delta x=0.1\).

Step 2 ：Euler's method is a numerical method used to approximate solutions to first order differential equations. It uses the idea of local linearity or linear approximation, where we use the tangent line at a known point to approximate the function at a new point.

Step 3 ：The general formula for Euler's method is: \(y_{n+1} = y_n + h*f(x_n, y_n)\) where h is the step size, f(x_n, y_n) is the derivative at the point (x_n, y_n), and (x_n, y_n) is the current point.

Step 4 ：In this case, we have: x = 1, y = 2, h = 0.1, target_x = 1.3.

Step 5 ：By applying Euler's method, we find that the approximate value of y when x=1.3 is 2.03.

Step 6 ：Final Answer: The approximate value of y when x=1.3 using Euler's method with increments of 0.1 is \(\boxed{2.03}\).