Which option best lists the four columns in the Euler's Method chart for a first-order ODE?

Which option best lists the four columns in the Euler's Method chart for a first-order ODE?

• A. (x,y) | Δx | dy/dx | dy
• B. (x,y) | Δt | dy/dt | dy
• C. (t) | Δx | dy/dx | dy
• D. (x,y) | Δx | dy/dt | dy

Answer: (A)

Euler's method for a first-order ODE in the form dy/dx = f(x, y) advances along the x-axis using the slope at the current point. At (x_i, y_i), you find the slope dy/dx = f(x_i, y_i). With a chosen step Δx, the vertical change is dy = (dy/dx) Δx, and you update to x_{i+1} = x_i + Δx and y_{i+1} = y_i + dy. So the natural chart lists the current state (x, y), the step size Δx, the slope dy/dx, and the resulting change dy. Using Δt and dy/dt would apply only if the equation were in terms of t rather than x, i.e., dy/dt, not dy/dx. The option that matches the dy/dx formulation with x as the independent variable is the correct one.

Euler's method for a first-order ODE in the form dy/dx = f(x, y) advances along the x-axis using the slope at the current point. At (x_i, y_i), you find the slope dy/dx = f(x_i, y_i). With a chosen step Δx, the vertical change is dy = (dy/dx) Δx, and you update to x_{i+1} = x_i + Δx and y_{i+1} = y_i + dy. So the natural chart lists the current state (x, y), the step size Δx, the slope dy/dx, and the resulting change dy. Using Δt and dy/dt would apply only if the equation were in terms of t rather than x, i.e., dy/dt, not dy/dx. The option that matches the dy/dx formulation with x as the independent variable is the correct one.