Question 1

Which of the following best describes a separable differential equation?

Accepted Answer

An equation that can be written as $$\frac{dy}{dx} = g(x)H(y)$$, where $$g(x) is a $$function of $$x$$ and $$H(y) is a $$function of $$y.$$

Question 2

Given the differential equation $$\frac{dy}{dx} = -2y$$, what is the general solution?

Accepted Answer

$$y = Ce$$^{-2x}$$

Question 3

Which of the following is the correct first step to solve $$\frac{dy}{dx} = x^2$ y as a $$separable equation?

Accepted Answer

$$\frac{1}{y} dy = x^2$ dx$$

Question 4

After separating variables in $$\frac{dy}{dx} = x^2$ y$$, what is the next step?

Accepted Answer

Integrate both sides: $$\int \frac{1}{y} dy = \int x^2$ dx$$

Question 5

What is the general solution to $$\frac{dy}{dx} = x^2$ y$$?

Accepted Answer

$$y = Ce$$^{\frac{x^3}$${3}}$$

Question 6

Given the initial value problem $$\frac{dy}{dx} = 2xy^2$, y(2$$) = \frac{1}{5}$$, what is the value of the constant C$ in the general solution y$$ = -\frac{1}{$$x^2$$} + C$$?

Accepted Answer

$$C = \frac{9}{4}$$

Question 7

Which of the following differential equations is linear?

Accepted Answer

$$x^2 y$$' + e^x y = 4$$

Question 8

For the equation $$\frac{dy}{dx} = \frac{6x^2$}{2y + \cos y}$$, which of the following is the correct separation of variables?

Accepted Answer

$$(2y + \cos y) dy = 6x^2$ dx$$

Question 9

What is the order of the differential equation $$\frac{d^2$y}{dx^2$} + 3\frac{dy}{dx} + 2y = 0$$?

Accepted Answer

Second order

Question 10

Which of the following is an implicit solution to $$\frac{dy}{dx} = x^2$ y$$?

Accepted Answer

$$\ln|y| = \frac{x^3$}{3} + C$$

Question 11

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 12

If $$y = Ce$$^{-2x}$$ is the general solution to y$$' = -2y$$, what is the particular solution if y(0) = 3$?

Accepted Answer

$$y = 3e$$^{-2x}$$

Question 13

Which of the following is NOT a characteristic of a linear differential equation?

Accepted Answer

It can have terms like $$y^3 or$$ $$\sin y.$$

Question 14

Suppose a population $$y$$ grows according to $$\frac{dy}{dx} = ky$$, where $$k \u003e 0. $$Which of the following best describes the long-term behavior of the population?

Accepted Answer

The population grows exponentially without bound.

Question 15

Which of the following is the correct integration step for $$\int \frac{1}{y} dy = \int x^2$ dx$$?

Accepted Answer

$$\ln|y| = \frac{x^3$}{3} + C$$

Question 16

A chemical reaction follows the law $$\frac{dy}{dt} = -ky$$, where $$y is $$the concentration of a reactant. What does the solution $$y = Ce$$^{-kt}$$ represent?

Accepted Answer

Exponential decay of the reactant concentration over time.

Separable Differential Equations and Initial Value Problems

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