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

- 10
- 8
- 6
- 5

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

- 0.54
- 0.44
- 0.64
- 0.66

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

- 1.00
- 1.03
- 0.97
- 0.96

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

- 0.01
- 0.031
- 0.0631
- 0.1

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
- τ/2
- τ
- 2τ

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?

- 1.33
- 1.67
- 2.00
- 2.33