Applications of First-Order PDEs: Interactive Learning Quizzes

Question 1

Which field does NOT employ first-order partial differential equations (PDEs)?

Accepted Answer

Quantum mechanics

Answer

Fluid dynamics

Answer

Heat transfer

Answer

Acoustics

Question 2

Which of the following is NOT an application of first-order partial differential equations in the field of physics?

Accepted Answer

Quantum field theory

Answer

Fluid mechanics

Answer

Heat transfer

Question 3

Identify the first-order partial differential equation among the following:

Accepted Answer

y(∂z/∂x) + x(∂z/∂y) = z

Answer

x(∂^2z/∂x^2) + y(∂^2z/∂y^2) = z

Answer

(∂^2z/∂x^2) + (∂^2z/∂y^2) = 0

Answer

∫zdxdy = xy

Question 4

In fluid mechanics, the equation governing the velocity potential φ is ∂^2φ/∂x^2 + ∂^2φ/∂y^2 = 0. What type of PDE is this?

Accepted Answer

Laplace equation

Answer

Heat equation

Answer

Poisson equation

Answer

Wave equation

Question 5

Which boundary condition is suitable for solving the heat equation in a bounded domain?

Accepted Answer

Dirichlet boundary condition

Answer

Mixed boundary condition

Answer

Cauchy boundary condition

Answer

Periodic boundary condition

Question 6

The acoustics equation is given by: ∂^2u/∂t^2 - c^2∇^2u = 0. What does this equation describe?

Accepted Answer

Wave propagation

Answer

Heat transfer

Answer

Convection

Answer

Diffusion

Question 7

Which numerical method is commonly employed to solve two-dimensional first-order PDEs?

Accepted Answer

Finite difference method

Answer

Finite element method

Answer

Boundary element method

Question 8

Identify a nonlinear first-order PDE:

Accepted Answer

Burgers' equation

Answer

Diffusion equation

Answer

Laplace equation

Answer

Wave equation

Question 9

Which of the following is NOT an application of first-order partial differential equations?

Accepted Answer

Quantum mechanics

Answer

Fluid mechanics

Answer

Heat transfer

Answer

Acoustics

Question 10

In the wave equation ∂²u/∂t² = c²∇²u, what does the constant c represent?

Accepted Answer

Wave speed

Answer

Wavelength

Answer

Amplitude

Answer

Frequency

Question 11

Which numerical method is commonly employed to solve first-order PDEs?

Accepted Answer

Finite difference method

Answer

Monte Carlo method

Answer

Support vector machine

Answer

Genetic algorithm

Question 12

In a compressible fluid flow, the continuity equation ∂ρ/∂t + ∇·(ρu) = 0 represents which physical principle?

Accepted Answer

Conservation of mass

Answer

Helmholtz theorem

Answer

Conservation of momentum

Answer

Stokes' theorem

Question 13

Which of the following is NOT a type of first-order PDE?

Accepted Answer

Second order PDE

Answer

Ultrahyperbolic equation

Answer

Parabolic equation

Answer

Hyperbolic equation

Question 14

Which of the following is the characteristic curve for the PDE u_t + uu_x = 0?

Accepted Answer

dx/dt = u

Answer

dx/dt = -u

Answer

du/dt = x

Answer

du/dt = -x

Question 15

A first-order partial differential equation (PDE) is given by ∂u/∂x + ∂u/∂y = 0. Which of the following functions u(x, y) satisfies this equation?

Accepted Answer

u(x, y) = x - y

Answer

u(x, y) = sin(x)cos(y)

Answer

u(x, y) = x^2 + y^2

Question 16

In acoustics, the pressure field p(x, y) satisfies the equation ∂²p/∂x² + ∂²p/∂y² = (1/c²)∂²p/∂t². Which of the following is a suitable boundary condition to ensure a standing wave solution?

Accepted Answer

p(x, y) = 0 on the boundaries

Answer

p(x, y) = p₀ on the boundaries

Answer

∂p/∂n = 0 on the boundaries

Question 17

Classify the following first-order quasilinear PDE: ∂u/∂x + u∂u/∂y = 0

Accepted Answer

Hyperbolic

Answer

Parabolic

Answer

Ultrahyperbolic

Answer

Elliptic

Question 18

The traffic flow on a highway is modeled by the equation ρt + (ρ(2-ρ))x = 0, where ρ(x,t) represents the density of cars. What is the physical interpretation of this equation?

Accepted Answer

The rate of change of car density at a point depends on the density and its gradient.

Answer

The car density is only a function of time.

Answer

The traffic flow is constant over time.

Question 19

A thin rod of length L with insulated ends has an initial temperature distribution given by f(x). Which of the following PDEs governs the temperature u(x,t) within the rod as a function of position x and time t?

Accepted Answer

∂u/∂t = k ∂²u/∂x²

Answer

∂²u/∂t² = k ∂²u/∂x²

Answer

∂u/∂t = k ∂u/∂x

Question 20

Which technique is not commonly used for solving first-order linear PDEs?

Accepted Answer

Separation of variables

Answer

Integrating factor method

Answer

Lagrange's method

Answer

Method of characteristics

Question 21

The wave equation describing the transverse vibrations of a string fixed at both ends is given by ∂²u/∂t² = c² ∂²u/∂x², where u(x,t) represents the displacement of the string. What type of PDE is this?

Accepted Answer

Second-order hyperbolic PDE

Answer

Second-order elliptic PDE

Answer

Second-order parabolic PDE

Answer

First-order hyperbolic PDE

Question 22

The equation ∂u/∂t + u ∂u/∂x = 0 models:

Accepted Answer

Inviscid fluid flow

Answer

Diffusion of a pollutant

Answer

Heat conduction in a rod

Answer

Wave propagation in a string

Question 23

The method of characteristics for solving first-order PDEs involves:

Accepted Answer

Finding curves along which the PDE reduces to an ODE.

Answer

Always leading to an explicit solution for the PDE.

Answer

Being applicable only to linear PDEs.

Question 24

A vibrating string is fixed at x = 0 and x = L. Its initial displacement is given by f(x) = sin(πx/L). What is the fundamental mode's spatial frequency?

Accepted Answer

π/L

Answer

2π/L

Answer

1/L

Answer

L/π

Question 25

Consider the first-order PDE $\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0$, where c is a constant. What is the general solution of this PDE?

Accepted Answer

$u(x,t) = f(x - ct)$

Answer

$u(x,t) = f(x) + g(t)$

Answer

$u(x,t) = f(ct - x)$

Answer

$u(x,t) = f(x + ct)$

Question 26

The diffusion equation $\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2}$ models the spread of a substance due to random motion. Which of the following factors does NOT directly influence the rate of diffusion according to this equation?

Accepted Answer

Initial concentration of the substance

Answer

Spatial gradient of the concentration

Answer

Diffusion coefficient (D)

Answer

Time elapsed

Question 27

A thin rod of length L has constant conductivity k. The ends of the rod are held at temperatures u(0, t) = 0 and u(L, t) = 100. Which of the following represents the steady-state temperature distribution along the rod?

Accepted Answer

u(x) = 100x/L

Answer

u(x) = 100(x/L)^2

Answer

u(x) = 100e^(-x/L)

Answer

u(x) = 100sin(πx/L)

Question 28

A fluid flows with velocity v = (y, -x) in a 2D space. If the concentration of a pollutant in the fluid is governed by the equation ∂u/∂x + ∂u/∂y = u, what is the expression for the concentration u?

Accepted Answer

u = e^x*e^y + f(y)

Answer

u = e^(-x)*e^(-y) - f(x)

Answer

u = e^(-x)*e^(-y) + f(y)

Answer

u = e^x*e^y - f(x)

Question 29

A thin wire is stretched from x = 0 to x = L and released at time t = 0. The displacement of the wire is governed by the wave equation ∂^2u/∂t^2 = c^2∂^2u/∂x^2, where c is the wave speed. Which of the following boundary conditions represents the fixed ends of the wire?

Accepted Answer

u(0, t) = 0, u(L, t) = 0

Answer

∂u/∂t(0, t) = 0, ∂u/∂t(L, t) = 0

Answer

∂u/∂x(0, t) = 0, ∂u/∂x(L, t) = 0

Answer

u(0, t) = 1, u(L, t) = 1

Question 30

A rectangular plate with sides a and b has boundary conditions u(0, y) = u(a, y) = 0 and u(x, 0) = f(x), u(x, b) = g(x). Which of the following methods can be used to solve for the steady-state temperature distribution on the plate?

Accepted Answer

Separation of Variables

Answer

Laplace Transform

Answer

Finite Element Method

Answer

Method of Characteristics

Question 31

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

Accepted Answer

u*∂u/∂x + y*∂u/∂y = 1

Answer

∂^2u/∂x^2 + ∂^2u/∂y^2 = 0

Answer

∂u/∂x + ∂u/∂y = u^2

Answer

x*∂u/∂x + y*∂u/∂y = u

Question 32

A projectile is fired from the origin with velocity (v, 0). The trajectory of the projectile is governed by the equation dy/dx = v/g(x), where g(x) is the acceleration due to gravity. Which of the following represents the equation of the trajectory?

Accepted Answer

y = vx - (1/2)gt^2

Answer

y = vx + (1/2)gt^2

Answer

y = (1/2)gt^2 - vx

Answer

y = (1/2)vx - gt^2

Question 33

Which of the following is a general solution to the equation ∂u/∂x + ∂u/∂y = 0?

Accepted Answer

u = f(x - y)

Answer

u = f(x + y)

Answer

u = f(x^2 + y^2)

Answer

u = f(xy)

Question 34

A thin membrane is stretched over a circular frame of radius R. The membrane is plucked and vibrates with a frequency f. Which of the following boundary conditions represents the clamped edge of the membrane?

Accepted Answer

u(R, θ, t) = 0

Answer

∂u/∂θ(R, θ, t) = 0

Answer

u(0, θ, t) = 0

Answer

∂u/∂r(R, θ, t) = 0

Question 35

A fluid flows past a cylinder with radius a. The velocity of the fluid far from the cylinder is U. Which of the following represents the stream function for the flow?

Accepted Answer

ψ = Ur(1 - a^2/r^2)sinθ

Answer

ψ = Ur(1 - a^2/r^2)cosθ

Answer

ψ = Ur(1 + a^2/r^2)sinθ

Answer

ψ = Ur(1 + a^2/r^2)cosθ

Question 36

A vibrating string with fixed ends has a fundamental frequency of 100 Hz. If the string is shortened by 25%, what is its new fundamental frequency?

Accepted Answer

125 Hz

Answer

50 Hz

Answer

75 Hz

Answer

150 Hz

Question 37

A thin rod of length L has an initial temperature distribution given by f(x). Its ends are kept at 0 degrees Celsius for all time t > 0. Which of the following PDEs describes the temperature u(x, t) of the rod?

Accepted Answer

∂u/∂t = k(∂²u/∂x²)

Answer

∂u/∂t = k(∂u/∂x)

Answer

∂²u/∂t² = k(∂²u/∂x²)

Answer

∂u/∂t = k(∂³u/∂x³)

Question 38

The traffic flow on a highway is modeled by the equation ρt + (ρv)x = 0, where ρ is the density of cars and v is their velocity. What does this equation represent physically?

Accepted Answer

Conservation of cars: the rate of change of the number of cars in any segment of the highway equals the difference between the inflow and outflow of cars.

Answer

Newton's second law of motion applied to the traffic flow.

Question 39

Which method is commonly used to solve the following first-order PDE:  a(x,y) ∂u/∂x + b(x,y) ∂u/∂y = c(x,y)?

Accepted Answer

Method of characteristics

Answer

Separation of variables

Answer

Laplace transform

Answer

Fourier transform

Question 40

A vibrating string is fixed at both ends. The displacement u(x,t) of the string satisfies the wave equation ∂²u/∂t² = c² ∂²u/∂x², where c is the wave speed. What do the boundary conditions u(0,t) = 0 and u(L,t) = 0 represent physically?

Accepted Answer

The ends of the string are fixed and do not move.

Answer

The initial velocity of the string is zero.

Answer

The wave speed is zero at the ends of the string.

Answer

The initial displacement of the string is zero.

Question 41

A pollutant is spreading in a river according to the advection-diffusion equation ∂C/∂t = D ∂²C/∂x² - v ∂C/∂x, where C is the concentration of the pollutant, D is the diffusion coefficient, and v is the river's velocity. What is the effect of increasing the diffusion coefficient D on the spread of the pollutant?

Accepted Answer

The pollutant spreads more rapidly.

Answer

The pollutant spreads at a constant rate.

Answer

The pollutant spreads less rapidly.

Answer

The diffusion coefficient has no effect on the spread of the pollutant.

Question 42

What is a characteristic curve in the context of solving first-order PDEs?

Accepted Answer

A curve in the solution domain along which the PDE reduces to an ODE.

Answer

A curve that shows the initial condition of the PDE.

Answer

A curve that represents the boundary of the solution domain.

Question 43

The equation ∂u/∂t + u ∂u/∂x = 0 is an example of which type of first-order PDE?

Accepted Answer

Nonlinear advection equation

Answer

Heat equation

Answer

Linear advection equation

Answer

Wave equation

Question 44

In the context of fluid mechanics, what does the continuity equation ∂ρ/∂t + ∇ ⋅ (ρv) = 0 represent?

Accepted Answer

Conservation of mass: the rate of change of mass in a control volume equals the net mass flux across its boundaries.

Answer

Conservation of momentum

Question 45

When using the method of characteristics to solve a first-order PDE, what is the significance of a shock wave formation?

Accepted Answer

It indicates that the solution becomes multi-valued, and a discontinuity arises.

Answer

It implies that the PDE has no solution.

Answer

It suggests that the method of characteristics is not applicable.

Question 46

Consider the traffic flow equation ρt + (ρv)x = 0. If the velocity v is constant, what can be said about the solution ρ(x,t)?

Accepted Answer

The solution represents a wave traveling with velocity v.

Answer

The solution always decays to zero.

Answer

The solution is always constant.