Quiz Chapter 08.04: Runge-Kutta 4th Order Method


(All Tests)


(More on Runge-Kutta 4th Order Method)


(More on Ordinary Differential Equations)

Pick the most appropriate answer

1. To solve the ordinary differential equation 3 \dfrac{dy}{dx} + xy^{2} = \sin x, \, y(0) = 5, by Runge-Kutta 4th order method, you need to rewrite the equation as


2. Given

      • 3 \dfrac{dy}{dx} + 5y^{2} = \sin x, \, y(0.3) = 5

and using a step size of h=0.3, the value of y(0.9) using Runge-Kutta 4th order method is most nearly


3. Given

      • 3 \dfrac{dy}{dx} + y^{2} = e^{x}, \, y(0.3) = 5

and using a step size of h=0.3, the best estimate of \dfrac{dy}{dx} (0.9) using Runge-Kutta 4th order method is most nearly


4. The velocity (m/s) of a parachutist is given as a function of time (seconds) by

      • v(t) = 55.8 \tanh (0.17t), \, t \geq 0

Using Runge-Kutta 4th order method with a step size of 5 seconds, the distance traveled by the body from t=2 to t=12 seconds is estimated most nearly as


5. Runge-Kutta method can be derived from using the first three terms of Taylor series of writing the value of y_{i+1}, that is the value of y at x_{i+1}, in terms of y_{i} and all the derivatives of y at x_{i}. If h=x_{i+1} - x_{i}, the explicit expression for y_{i+1} if the first five terms of the Taylor series are chosen for the ordinary differential equation

      • \dfrac{dy}{dx} + 5y = 3e^{-2x}, \, y(0) = 7

would be


6. A hot, solid cylinder is immersed in a cool oil bath as part of a quenching process. This process makes the temperature of the cylinder, \theta_{c}, and the bath, \theta_{b}, change with time.

If the initial temperature of the bar and the oil bath is given as 600^{\circ}C and 27^{\circ}C, respectively, and:

      • radius of cylinder =3 cm
      • density of cylinder =2700 kg/m3
      • specific heat of cylinder =895 J/(kg-K)
      • convection heat transfer coefficient =100 W/(m2-K)
      • specific heat of oil =1910 J/(kg-K)
      • mass of oil =2 kg

The coupled ordinary differential equations governing the heat transfer are given by