Q: A tank initially contains 100 liters of pure water. Saltwater containing 2 grams of salt per liter flows in at a rate of 3 liters per minute, and the well-mixed solution flows out at the same rate. Let $$Q(t) be $$the amount of salt in the tank at time $$t (in $$minutes). Which differential equation models this scenario?

$$\frac{dQ}{dt} = 2 \times 3 - \frac{3}{100} Q$$

Question 1

Which of the following best describes a differential equation?

Accepted Answer

An equation that relates a function to its derivatives.

Question 2

A tank initially contains 100 liters of pure water. Saltwater containing 2 grams of salt per liter flows in at a rate of 3 liters per minute, and the well-mixed solution flows out at the same rate. Let $$Q(t) be $$the amount of salt in the tank at time $$t (in $$minutes). Which differential equation models this scenario?

Accepted Answer

$$\frac{dQ}{dt} = 2 	imes 3 - \frac{3}{100} Q$$

Question 3

Which of the following is a system of differential equations?

Accepted Answer

$$\frac{dx}{dt} = 2x + y, \quad \frac{dy}{dt} = x - y$$

Question 4

Consider the linear system $$\frac{dx}{dt} = 3x + 4y$$, $$\frac{dy}{dt} = -4x + 3y. $$What type of equilibrium point does the origin represent?

Accepted Answer

A center

Question 5

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

Accepted Answer

$$y'' + 5y' + 6y = 0$$

Question 6

A mass-spring-damper system is modeled by $$m y'' + c y' + k y = 0. If$$ $$m = 1 kg$$, $$c = 2 Ns/m$$, and $$k = 5 N/m$$, what is the characteristic equation?

Accepted Answer

$$r^2 + 2r + 5 = 0$

Question 7

Which of the following systems is nonlinear?

Accepted Answer

$$\frac{dx}{dt} = x^2$ + y$$

Question 8

The Laplace transform of $$f(t) = e$$^{2t}$$ is:

Accepted Answer

$$\frac{1}{s-2}$$

Question 9

Which property of the Laplace transform is used to solve initial value problems involving derivatives?

Accepted Answer

$$\mathcal{L}\{f'(t)\} = sF(s) - f(0)$$

Question 10

A predator-prey model is given by $$\frac{dx}{dt} = x(1 - y)$$, $$\frac{dy}{dt} = -y(1 - x). $$Which of the following points is an equilibrium?

Accepted Answer

$$(1, 1)$$

Question 11

Which of the following statements about linear systems is TRUE?

Accepted Answer

The principle of superposition applies to linear systems.

Question 12

Given the Laplace transform $$F(s) = \frac{2}{s^2$ + 4}$$, what is the original function $$f(t)$$?

Accepted Answer

$$f(t) = \sin(2t)$$

Question 13

Which of the following best describes a homogeneous system of differential equations?

Accepted Answer

All terms involve the dependent variables or their derivatives, and there are no constant or forcing terms.

Question 14

A circuit with a resistor, inductor, and capacitor in series is modeled by $$L\frac{d^2$q}{dt^2$} + R\frac{dq}{dt} + \frac{1}{C}q = E(t). If$$ $$L=1 H$$, $$R=0$$, $$C=1 F$$, and $$E(t)=0$$, what is the general solution for $$q(t)$$?

Accepted Answer

$$q(t) = A\cos t + B\sin t$$

Question 15

Which of the following is NOT a property of the Laplace transform?

Accepted Answer

It always preserves the time domain.

Question 16

Suppose $$\frac{dx}{dt} = 2x$$, $$x(0) = 5. $$What is the solution for $$x(t)$$?

Accepted Answer

$$x(t) = 5e$$^{2t}$$

Question 17

Which of the following is a real-world application of a system of differential equations?

Accepted Answer

Modeling the interaction between two competing species in an ecosystem.

Question 18

Given the nonlinear system $$\frac{dx}{dt} = x - xy$$, $$\frac{dy}{dt} = -y + xy$$, what is the equilibrium point where both populations coexist?

Accepted Answer

$$(1, 1)$$

Question 19

Which Laplace transform property allows you to solve differential equations with shifted inputs, such as $$f(t-a)u(t-a)$$?

Accepted Answer

$$\mathcal{L}\{f(t-a)u(t-a)\} = e$$^{-as}$$F(s)$$

Study Notes on Differential Equations and Systems

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