BC Unit 7 Jeopardy

Differential Equations

Euler's Method

100

A population described by a function P at time t decreases at a rate proportional to P. Which of the following could be the differential equation for the rate of change of the population?

A. dP/dt = -0.015P
B. dP/dt = -0.015/P
C. dP/dt = -0.02t
D. dP/dt = 6e^-0.02t

A. dP/dT = -0.015P

100

Let y = f(x) be the solution to the differential equation dy/dx = x - y with initial condition f(1) = 3. What is the approximation for f(2) obtained by using Euler's Method with two steps of equal length starting at x = 1?

f(2) ≅ 7/4

200

The rate at which a quantity M of a certain radioactive substance decays is proportional to the amount of the substance present at a given time. Which of the following is a differential equation that could describe this relationship?

A. dM/dt = -3.7t²
B. dM/dt = -0.11M
C. dM/dt = 0.08t²
D. dM/dt = 1.2M

B. dM/dt = -0.11M

200

Let y = f(x) be the solution to the differnetial equation dy/dx = 2y - x with initial condition f(1) = 2. What is the approximation for f(0) using Euler's Method with 2 steps* starting at x = 1?

f(0) ≅ 1/4

300

A vehicle moves along a straight road. The vehicle's position is given by f(t), where t is measured in seconds since the vehicle starts moving. During the first 10 seconds of the motion, the vehicle's acceleration is proportional to the cube root of the time since the start. Please write a differential equation with k as a positive constant.

d²f/dt² = k ³√t

300

x | 1.4 | 1.7 | 2.0 |

f'(x) | -8 | 2 | 3 |

The table above gives values of f' for selected values of x. If f(2) = 6, what is the approximation for f(1.4) using Euler's Method, starting at x=2 with two steps of equal size?

f(1.4) ≅ 4.5

400

A jogger runs along a straight track. The jogger's position is given by the function p(t), where t is measured in minutes since the start of the run. Please create a differential equation for this relationship, with k as a positive constant.

d²p/dt² = k√t

400

Let y = f(x) be the solution to the differential equation dy/dx = x + 2y with initial condition f(0) = 2. What is the approximation for f(-0.4) obtained using Euler's Method with two steps of equal length starting at x=0?

f(-0.4) ≅ 0.76

500

For what value of k, if any, will y = ke^-2x + 4cos(3x) be a solution to the differential equation y" + 9y = 26e^-2x?

k = 2.

(Explanation)

y = ke^-2x + 4cos(3x)
y' = -2ke^-2x - 12sin(3x)
y" = 4ke^-2x- 36cos(3x)

Then, y" + 9y = 4ke^-2x - 36cos(3x) + 9ke^-2x + 36cos(3x) = 13ke^-2x

If 13ke^-2x = 26e^-2x, then 13k = 26, k = 2.
Therefore, if k = 2, y=ke^-2x + 4cos(3x) is a solution to the differential equation y" + 9y = 26e^-2x.

500

Let y = f(x) be the solution to the differential equation dy/dx = u - 10x^2 with the initial condition f(0) = 3. What is the approximation for f(0.4) if Euler's Method is used, starting at x=0 with steps of size 0.2?

f(0.4) ≅ 4.240