Quiz Chapter 08.02: Euler’s Method for 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 9:00PM, 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