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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. A differential equation with a starting condition
C. A matrix equation with an eigenvalue
D. An algebraic equation with a boundary 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 calculate only the initial condition
B. To remove the independent variable
C. To approximate the solution values
D. To change the order of the equation

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 equation order
C. The step size
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 uses no step size
B. It uses an average slope
C. It avoids function evaluation
D. It replaces the differential equation

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

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

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

Runge-Kutta methods Easy
A. They require no function values
B. They use several slope evaluations
C. They use only the initial value
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. Three evaluations
B. Four evaluations
C. Five evaluations
D. Two 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 less accurate approximation
B. An exact symbolic solution
C. A more accurate approximation
D. A solution without computation

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

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

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

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

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

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

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

Multi-step methods Easy
A. An estimated next value
B. The exact initial value
C. The differential equation order
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 direct matrix method
C. A predictor-corrector method
D. A single-point interpolation method

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

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

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

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

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. must be a polynomial of degree two
B. and are continuous near
C. must be independent of
D. is continuous only in

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 always removes round-off errors completely
D. It usually improves accuracy but increases computation

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. Uses only the initial slope
B. Uses an average of slopes over the interval
C. Solves the equation symbolically
D. Eliminates the step-size requirement

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

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

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

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

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

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

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

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. is continuous in only
D. has a continuous second derivative in

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 computes two orders from shared stages
B. It requires a multistep history
C. It uses two independent meshes
D. It removes all truncation error

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. Second order
B. Fourth order
C. First order
D. Third order

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

Multi-step methods Hard
A. At least one root satisfies
B. All roots satisfy
C. All roots satisfy , with unit roots simple
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 initial derivative must equal the final derivative
B. Only is required
C. The first values require a one-step method
D. The method automatically generates all history values

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 has higher order with implicit stability
C. It is exact for every polynomial right-hand side
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. Its predictor may possess parasitic modes
B. Its interpolation polynomial is constant
C. Accuracy always implies instability
D. The corrector has no derivative terms

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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