# Quiz Chapter 08.02: Euler’s Method for Ordinary Differential Equations

 MULTIPLE CHOICE TEST EULER’S METHOD ORDINARY DIFFERENTIAL EQUATIONS

1. To solve the ordinary differential equation $3 \dfrac{dy}{dx} + 5y^{2} = \sin x, \, y(0) = 5$, by Euler’s 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 Euler’s method is most nearly

3. Given

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

and using a step size of $h=0.3$, the best estimate of $\dfrac{dy}{dx}(0.9)$ using Euler’s method is most nearly

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

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

Using Euler’s method with a step size of $5$ seconds, the distance in meters traveled by the body from $t=2$ to $t=12$ seconds is most nearly

5. Euler’s method can be derived by using the first two 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 $y_{i}$ 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 the Taylor series are chosen for the ordinary differential equation

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

would be

6. A homicide victim is found at $6$:$00$ PM in an office building that is maintained at $72^{\circ}F$. When the victim was found, his body temperature was at $85^{\circ}F$. Three hours later at $9$:$00$PM, his body temperature was recorded at $78^{\circ}F$. Assume the temperature of the body at the time of death is the normal human body temperature of $98.6^{\circ}F$.

The governing equation for the temperature $\theta$ of the body is

• $\dfrac{d \theta}{dt} = -k \left( \theta - \theta_{a} \right)$

where

• $\theta=$ temperature of the body, $^{\circ}F$
• $\theta_{a}=$ ambient temperature, $^{\circ}F$
• $t=$ time, hours
• $k=$ constant based on thermal properties of the body and air

The estimated time of death is most nearly