Quiz Chapter 08.03: Runge-Kutta 2nd Order Method

1. To solve the ordinary differential equation 3 \dfrac{dy}{dx} + xy^{2} = \sin x, \, y(0) = 5 by the Runge-Kutta 2^nd 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 the Runge-Kutta 2^nd order Heun’s method is most nearly

3. Given

• • • 3 \dfrac{dy}{dx} + 5 \sqrt{y} = e^{0.1x}, \, y(0.3) = 5,

and using a step size of h=0.3, the best estimate of \dfrac{dy}{dx} (0.9) using the Runge-Kutta 2^nd order midpoint-method is most nearly

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

• • • v(t) = 200 \ln (1+t) - t, \, t \geq 0

Using the Runge-Kutta 2^nd order Ralston method with a step size of 5 seconds, the distance in meters traveled by the body from t=2 and t=12 seconds is estimated most nearly as

5. The Runge-Kutta 2nd order method can be derived by using the first three terms of the 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} (that is the value of y at x_{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 three terms of Taylor series are chosen for solving the ordinary differential equation

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

would be

6. A spherical ball is taken out of a furnace at 1200K and is allowed to cool in air. Given the following:

• • • radius of ball =2 cm
    • specific heat of ball =420 J/(kg-K)
    • density of ball =7800 kg/m³
    • convection coefficient =350 J/(s-m²-K)

The ordinary differential equation is given for the temperature, \theta of the ball

• • • \dfrac{d \theta}{dt} = -2.20673 \times 10^{-13} \left( \theta^{4} - 81 \times 10^{8} \right)

if only radiation is accounted for. The ordinary differential equation if convection is accounted for in addition to radiation is