Euler Method MCQ

1. Consider the equation \frac{\mathrm{d} u}{\mathrm{d} t}=3t^2+1 with u = 0 at t = 0. This is numerically solved by using the forward Euler method with a step size Δt = 2. The absolute error in the solution in the end of the first time step is

  1. 10
  2. 8
  3. 6
  4. 5
Answer
Answer. b

2. The ordinary differential equation \frac{\mathrm{d} x}{\mathrm{d} t}=-3x+2, with x(0) = 1 is to be solved using the forward Euler method. The largest time step that can be used to solve the equation without making the numerical solution unstable is

  1. 0.54
  2. 0.44
  3. 0.64
  4. 0.66
Answer
Answer. d

3. While numerically solving the differential equation \frac{\mathrm{d} y}{\mathrm{d} x}+2xy^2=0, y(0) = 1 using Euler’s predictor-corrector (improved Euler-Cauchy) with a step size of 0.2, the value of y after the first step is

  1. 1.00
  2. 1.03
  3. 0.97
  4. 0.96
Answer
Answer. d

4. Consider a differential equation \frac{\mathrm{d} y(x)}{\mathrm{d} x}-y(x)=x with the initial condition y(0) = 0. Using Euler’s first order method with a step size of 0.1, the value of y(0.3) is

  1. 0.01
  2. 0.031
  3. 0.0631
  4. 0.1
Answer
Answer. b

5. The differential equation \frac{\mathrm{d} x}{\mathrm{d} t}=\frac{1-x}{\tau} is discretised using Euler’s numerical integration method with a time step ΔT > 0. What is the maximum permissible value of ΔT to ensure stability of the solution of the corresponding discrete-time equation?

  1. 1
  2. τ/2
  3. τ
  4. 2τ
Answer
Answer. d

6. The differential equation \frac{\mathrm{d} y}{\mathrm{d} x}=0.25y^2 is to be solved using the backward (implicit) Euler’s method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?

  1. 1.33
  2. 1.67
  3. 2.00
  4. 2.33
Answer
Answer. c