Q: Given $$y'(t) = 2.5y(t)$$ and $$y(0) = 5$$, what is $$y(2)$$?

$$5e^{2.5 \times 2}$$ = $$5e^{5}$$

Question 1

Which of the following best defines a differential equation?

Accepted Answer

An equation that relates an unknown function to one or more of its derivatives.

Question 2

Given the differential equation $$\frac{dy}{dt} = -6.2y$$, what type of differential equation is this?

Accepted Answer

First-order differential equation

Question 3

Which of the following is a second-order differential equation?

Accepted Answer

$$y'' + 2y' + y = 4$$

Question 4

Suppose $$y(t) = Ce$$^{2.5t}$$ is a solution to y$$'(t) = 2.5y(t)$$. What is y$$'(t)$$?

Accepted Answer

$$2.5Ce^{2.5t}$$

Question 5

Given the initial value problem $$y'(t) = 2.5y(t)$$ and $$y(0) = 3.2$$, what is the particular solution?

Accepted Answer

$$y(t) = 3.2e$$^{2.5t}$$

Question 6

Which of the following functions is NOT a solution to $$y'' - 4y = 0$$?

Accepted Answer

$$y(x) = e$$^{x}$$

Question 7

A population of bacteria grows according to the differential equation $$\frac{dN}{dt} = kN. If$$ $$N(0) = 100$$ and $$k = 0.03$$, what is the population at time $$t$$?

Accepted Answer

$$N(t) = 100e$$^{0.03t}$$

Question 8

Which of the following statements is TRUE about the solution $$y(t) = Ce$$^{2.5t}$$ to y$$'(t) = 2.5y(t)$$?

Accepted Answer

The constant $$C$$ can be determined if an initial value is given.

Question 9

If $$y(x) = 2e$$^{2x}$$, what is y$$''(x) + y'(x)$$?

Accepted Answer

$$8e^{2x}$$ + $$4e^{2x}$$ = $$12e^{2x}$$

Question 10

Which of the following best describes an initial value problem?

Accepted Answer

A differential equation together with a specified value of the unknown function at a given point.

Question 11

Suppose $$y(x) = e$$^{2x}$$ and y$$'' - 4y = 0$$. Is y(x$$)$$ a solution to the equation?

Accepted Answer

Yes, because $$y'' = 4e$$^{2x}$$ and 4e$$^{2x}$$ - 4e$$^{2x}$$ = 0.$$

Question 12

A radioactive substance decays according to $$\frac{dN}{dt} = -kN. If$$ $$N(0) = N_0$, what is the general solution?

Accepted Answer

$$N(t) = N_0e$$^{-kt}$$

Question 13

Which of the following is a first-order differential equation?

Accepted Answer

$$y' = -y$$

Question 14

If $$y(t) = Ce$$^{2.5t}$$ is a solution to y$$'(t) = 2.5y(t)$$, what is the value of C$ if y(0) = 10$?

Accepted Answer

$$C = 10$$

Question 15

Which of the following real-world scenarios can be modeled by the differential equation $$\frac{dN}{dt} = -kN$$?

Accepted Answer

Radioactive decay of a substance

Question 16

Given $$y'(t) = 2.5y(t)$$ and $$y(0) = 5$$, what is $$y(2)$$?

Accepted Answer

$$5e^{2.5 	imes 2}$$ = $$5e^{5}$$

Question 17

Which of the following is the general solution to $$y'(t) = ky(t)$$?

Accepted Answer

$$y(t) = Ce$$^{kt}$$

Question 18

If $$y(x) = e$$^{2x}$$, what is y$$''(x)$$?

Accepted Answer

$$4e^{2x}$$

Introduction to Differential Equations

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