Quiz Chapter 08.03: Runge-Kutta 2nd Order Method

 MULTIPLE CHOICE TEST RUNGE-KUTTA 2nd ORDER METHOD ORDINARY DIFFERENTIAL EQUATIONS

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 $2$nd 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 $1200$K 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