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Unit 4: Numerical Solution of Ordinary Differential Equations - Practice Quiz

ECE183 — Mathematics For Robotics 60 Questions
0 Correct 0 Wrong 60 Left
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1 What is an initial value problem for an ordinary differential equation?

Numerical methods for initial value problem Easy
A. A difference equation without a starting value
B. An algebraic equation with a boundary condition
C. A matrix equation with an eigenvalue
D. A differential equation with a starting condition

2 For the problem , which value is usually known at the initial point?

Numerical methods for initial value problem Easy
A.
B.
C.
D.

3 What is the main purpose of a numerical method for an ordinary differential equation?

Numerical methods for initial value problem Easy
A. To remove the independent variable
B. To change the order of the equation
C. To approximate the solution values
D. To calculate only the initial condition

4 Which formula represents Euler's method for ?

Euler and Modified Euler method Easy
A.
B.
C.
D.

5 In Euler's method, what does usually represent?

Euler and Modified Euler method Easy
A. The final value
B. The step size
C. The equation order
D. The initial slope

6 Euler's method uses which slope to estimate the next value?

Euler and Modified Euler method Easy
A. The slope at the final point
B. The slope at the initial point only
C. The average of all past slopes
D. The slope at the current point

7 What is the main improvement in the Modified Euler method?

Euler and Modified Euler method Easy
A. It avoids function evaluation
B. It uses no step size
C. It replaces the differential equation
D. It uses an average slope

8 The Modified Euler method is also commonly called which method?

Euler and Modified Euler method Easy
A. The Jacobi method
B. The Heun method
C. The Newton method
D. The Gauss method

9 What is the main feature of Runge-Kutta methods?

Runge-Kutta methods Easy
A. They use only the initial value
B. They use several slope evaluations
C. They require no function values
D. They solve only linear equations

10 How many slope evaluations are used in the classical fourth-order Runge-Kutta method?

Runge-Kutta methods Easy
A. Five evaluations
B. Two evaluations
C. Four evaluations
D. Three evaluations

11 Which symbol commonly denotes the first slope in the RK4 method?

Runge-Kutta methods Easy
A.
B.
C.
D.

12 Compared with the basic Euler method, RK4 generally provides which type of result?

Runge-Kutta methods Easy
A. A solution without computation
B. A more accurate approximation
C. An exact symbolic solution
D. A less accurate approximation

13 What is a defining feature of a multi-step method?

Multi-step methods Easy
A. It uses values from previous steps
B. It uses only the final value
C. It uses no derivative values
D. It uses only the current value

14 Which method is an example of a multi-step method?

Multi-step methods Easy
A. Adams-Bashforth method
B. Bisection method
C. Newton-Raphson method
D. Euler method

15 What is usually required to start a multi-step method?

Multi-step methods Easy
A. A zero step size
B. Only the final boundary value
C. An algebraic factorization
D. Starting values from a one-step method

16 A predictor formula in a predictor-corrector method is used to obtain what?

Multi-step methods Easy
A. The differential equation order
B. An estimated next value
C. The exact initial value
D. The step size automatically

17 The Milne method is generally classified as which type of numerical method?

Milne method Easy
A. A root-finding method
B. A predictor-corrector method
C. A single-point interpolation method
D. A direct matrix method

18 How many previous points are commonly used in the Milne predictor formula?

Milne method Easy
A. Four previous points
B. One previous point
C. Six previous points
D. Two previous points

19 What is the purpose of the corrector step in the Milne method?

Milne method Easy
A. To improve the predicted value
B. To change the independent variable
C. To remove the initial condition
D. To increase the step size

20 Which expression represents Lagrange's equation for a mechanical system?

Lagrange's equation Easy
A.
B.
C.
D.

21 For the initial value problem , , which condition is primarily required to ensure a unique solution near ?

Numerical methods for initial value problem Medium
A. is continuous only in
B. must be a polynomial of degree two
C. and are continuous near
D. must be independent of

22 For a numerical solution of an initial value problem using step size , what is the main effect of reducing ?

Numerical methods for initial value problem Medium
A. It changes the differential equation being solved
B. It decreases accuracy because fewer points are used
C. It usually improves accuracy but increases computation
D. It always removes round-off errors completely

23 Using Euler's method with for , , the approximate value of is:

Euler and Modified Euler method Medium
A.
B.
C.
D.

24 For , , and , what is the predictor value obtained by Euler's method for ?

Euler and Modified Euler method Medium
A.
B.
C.
D.

25 For , , and , the modified Euler method using the trapezoidal correction gives approximately

Euler and Modified Euler method Medium
A.
B.
C.
D.

26 Compared with the basic Euler method, the modified Euler method is generally more accurate because it:

Euler and Modified Euler method Medium
A. Solves the equation symbolically
B. Eliminates the step-size requirement
C. Uses an average of slopes over the interval
D. Uses only the initial slope

27 For the classical fourth-order Runge-Kutta method, how many slope evaluations are made during one step?

Runge-Kutta methods Medium
A. Three evaluations
B. Four evaluations
C. Two evaluations
D. One evaluation

28 In the classical RK4 method, the weighted average of slopes used to update is:

Runge-Kutta methods Medium
A.
B.
C.
D.

29 For , , and , the RK4 approximation to is closest to:

Runge-Kutta methods Medium
A.
B.
C.
D.

30 The classical fourth-order Runge-Kutta method has a local truncation error of order:

Runge-Kutta methods Medium
A.
B.
C.
D.

31 A major feature of a linear multi-step method is that it:

Multi-step methods Medium
A. Uses only the current point
B. Requires no starting values
C. Always has fourth-order accuracy
D. Uses several previously computed solution values

32 Which formula represents the two-step Adams-Bashforth explicit method?

Multi-step methods Medium
A.
B.
C.
D.

33 Why are starting values needed when applying a three-step method?

Multi-step methods Medium
A. The differential equation has three unknown variables
B. The step size must be selected three times
C. The method cannot evaluate the derivative
D. The method requires values at three previous points

34 In predictor-corrector multi-step methods, the predictor is mainly used to:

Multi-step methods Medium
A. Determine the exact solution
B. Replace the initial condition
C. Reduce the number of known values
D. Estimate the next value explicitly

35 The Milne predictor formula for equally spaced points is:

Milne method Medium
A.
B.
C.
D.

36 The Milne corrector formula is commonly written as:

Milne method Medium
A.
B.
C.
D.

37 Before applying the Milne method, the required initial values are usually generated using:

Milne method Medium
A. The Milne corrector alone
B. Only the final boundary condition
C. A one-step method such as RK4
D. A random selection of mesh values

38 For a mechanical system with generalized coordinate , Lagrange's equation is:

Lagrange's equation Medium
A.
B.
C.
D.

39 For a simple pendulum with angle , mass , length , and gravity , the equation obtained from Lagrange's equation is:

Lagrange's equation Medium
A.
B.
C.
D.

40 For a conservative mechanical system, the Lagrangian is defined as:

Lagrange's equation Medium
A.
B.
C.
D.

41 For the initial-value problem , which condition is sufficient to guarantee local existence and uniqueness of the solution near ?

Numerical methods for initial value problem Hard
A. is continuous and locally Lipschitz in
B. is bounded in a neighborhood
C. has a continuous second derivative in
D. is continuous in only

42 A one-step method has local truncation error . Assuming the method is stable for the given IVP, what is its expected global discretization error?

Numerical methods for initial value problem Hard
A.
B.
C.
D.

43 For the test equation , a numerical method has amplification factor , where . Which condition characterizes absolute stability at a particular step size?

Numerical methods for initial value problem Hard
A. for every real
B. for every complex
C. independently of
D. for the selected

44 Applying explicit Euler to gives . For which interval of positive step sizes is the method absolutely stable?

Euler and Modified Euler method Hard
A.
B.
C.
D.

45 Using Heun's modified Euler method with predictor and corrector , compute one step for , , and .

Euler and Modified Euler method Hard
A.
B.
C.
D.

46 For the linear test equation , the explicit midpoint modified Euler method has amplification factor . Along the negative real axis, which condition describes its absolute-stability interval?

Euler and Modified Euler method Hard
A.
B.
C.
D.

47 For the classical fourth-order Runge-Kutta method applied to , which stability polynomial is obtained?

Runge-Kutta methods Hard
A.
B.
C.
D.

48 Apply classical RK4 with to , . Which approximation is obtained for ?

Runge-Kutta methods Hard
A.
B.
C.
D.

49 Which property distinguishes an embedded Runge-Kutta pair from a single Runge-Kutta formula?

Runge-Kutta methods Hard
A. It uses two independent meshes
B. It computes two orders from shared stages
C. It removes all truncation error
D. It requires a multistep history

50 A Runge-Kutta method is claimed to have order . Which condition must necessarily hold for its stability polynomial near ?

Runge-Kutta methods Hard
A.
B.
C.
D.

51 The two-step Adams-Bashforth method is . What is its order of accuracy?

Multi-step methods Hard
A. Third order
B. Fourth order
C. Second order
D. First order

52 For a linear multistep method with first characteristic polynomial , what is the root condition for zero-stability?

Multi-step methods Hard
A. All roots satisfy , with unit roots simple
B. All roots satisfy
C. At least one root satisfies
D. All roots satisfy , with unit roots repeated

53 Which statement correctly describes the practical startup of a -step explicit multistep method for an IVP?

Multi-step methods Hard
A. The method automatically generates all history values
B. The first values require a one-step method
C. Only is required
D. The initial derivative must equal the final derivative

54 For the Adams-Moulton corrector , what is its principal advantage over the two-step Adams-Bashforth formula?

Multi-step methods Hard
A. It requires no previous solution values
B. It is exact for every polynomial right-hand side
C. It has higher order with implicit stability
D. It is explicit and uses fewer evaluations

55 The Milne predictor formula is . Given , , , , and , compute the predicted value.

Milne method Hard
A.
B.
C.
D.

56 The Milne corrector is . If , , , , and , what is the corrected value?

Milne method Hard
A.
B.
C.
D.

57 Why can the Milne predictor-corrector method require a stability check even when its local truncation error is small?

Milne method Hard
A. The corrector has no derivative terms
B. Accuracy always implies instability
C. Its predictor may possess parasitic modes
D. Its interpolation polynomial is constant

58 For generalized coordinates , kinetic energy , potential energy , and generalized nonconservative force , which equation is Lagrange's equation?

Lagrange's equation Hard
A.
B.
C.
D.

59 For a single-link pendulum with inertia , mass , center-of-mass distance , gravity , and applied torque , which equation follows from Lagrange's formulation?

Lagrange's equation Hard
A.
B.
C.
D.

60 For a robot manipulator modeled by , which structural form is obtained after applying Lagrange's equations?

Lagrange's equation Hard
A.
B.
C.
D.
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