Find the exact solution to the initial value problem and

Question

Given that dy/dt = y + e^t with y(0) = 0.

i) Apply Euler's Method to find y_a (1) using step size h = 0.1;

ii) Apply Improved Euler's Method to find y_b (1) using step size h = 0.1;

iii) Find the exact solution to the initial value problem;

iv) Estimate the errors in the predicted value of y_a(1) and yb (1) and suggest a value of step size for each method that would provide a value of y(1) accurate to 0.5%.