For this question, please leave out the proof. Assume the formula is already known.
Also, instead of assuming a unit step of U(t) = 1(t), change the question to a sinusoidal input of U(t) = sin(2t).
Please solve this question and explain all the steps in detail.
For the equation you had provided for the modified Euler formula, I have now figured out what the general form of the differential equation is:
dx/dt = ax + bu = f(x,t)
Then the modified Euler formula is,
The solution provides a calculation involving Euler's formula as well as a simple proof for the formula.