Differential Equations for Engineering: Master the Fundamentals

Question 1

Which of the following is not a type of differential equation?

Accepted Answer

Algebraic equation

Answer

Ordinary differential equation

Answer

Partial differential equation

Answer

Linear differential equation

Question 2

Classify the differential equation y'' + 2y' + y = 0 based on its order and homogeneity.

Accepted Answer

Second order, homogeneous

Answer

First order, homogeneous

Answer

Second order, non-homogeneous

Answer

First order, non-homogeneous

Question 3

Which of the following is NOT a type of first order differential equation?

Accepted Answer

Partial

Answer

Linear

Answer

Nonlinear

Answer

Autonomous

Question 4

Classify the differential equation:

y'' + 2y' + 5y = 0

Is it a first or second order, and linear or non-linear differential equation?

Accepted Answer

Second order, linear

Answer

First order, linear

Answer

First order, non-linear

Answer

Second order, non-linear

Question 5

Solve the second-order differential equation and provide the general solution: $$y'' - 2y' + y = 0$$

Accepted Answer

$$y = c_1 e^x + c_2 e^{-x}$$

Answer

$$y = c_1 + c_2 e^x$$

Answer

$$y = c_1 sin x + c_2 cos x$$

Answer

$$y = c_1 x + c_2$$

Question 6

In engineering practice, which of the following applications does NOT involve the use of differential equations?

Accepted Answer

Calculating depreciation schedules for financial assets

Answer

Modeling the oscillations of a pendulum

Answer

Simulating the flow of heat in a solid object

Answer

Designing optimal control systems for machines

Question 7

Which of the following is not a type of differential equation?

Accepted Answer

Algebraic equation

Answer

Ordinary differential equation

Answer

Partial differential equation

Answer

Integral equation

Question 8

In the differential equation y'' + 2ay' + (a^2 + 1)y = 0, where 'a' is a constant, for which value(s) of 'a' will the corresponding solutions oscillate?

Accepted Answer

a < 0

Answer

a = 0

Answer

a > 0

Question 9

Which differential equation accurately describes the relationship between acceleration (a), velocity (v), and displacement (x) in a damped mechanical system?

Accepted Answer

ma + bv + kx = 0

Answer

F = ma

Answer

F = kx

Question 10

When solving the differential equation $y''-y=xe^x$ using the method of undetermined coefficients, which of the following is an appropriate form for the trial solution?

Accepted Answer

y = (Ax+B)e^x

Answer

y = Ae^x + Be^-x

Answer

y = Ae^x + Be^x + Ce^-x

Question 11

When faced with a Cauchy-Euler differential equation, where the coefficients are polynomials in the independent variable x, which method is typically used to find its solution?

Accepted Answer

Solving the characteristic equation

Answer

Method of undetermined coefficients

Answer

Variation of parameters

Answer

Series solutions

Question 12

In a spring-mass-damper system, the equation of motion is: m*x''(t) + c*x'(t) + k*x(t) = F(t). What is the order of this differential equation?

Accepted Answer

Second order

Answer

First order

Answer

Third order

Answer

Fourth order

Question 13

What is the degree of a partial differential equation involving derivatives up to the second order with respect to two independent variables?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 14

In solving linear second-order homogeneous differential equations with constant coefficients, which method is commonly employed to find the general solution?

Accepted Answer

Characteristic Equation Method

Answer

Method of Variation of Parameters

Answer

Method of Undetermined Coefficients

Question 15

Identify the type of partial differential equation given by: $$ rac{partial u}{partial t} + rac{partial u}{partial x} - u rac{partial u}{partial y} = 0 $$.

Accepted Answer

First-order quasi-linear partial differential equation

Answer

Second-order linear partial differential equation

Answer

First-order linear partial differential equation

Question 16

Identify the type of differential equation represented by: y'' + 2y' + 5y = e^x

Accepted Answer

Second-order linear differential equation with constant coefficients

Answer

First-order linear differential equation

Answer

Nonlinear differential equation

Answer

Partial differential equation

Question 17

Determine whether the differential equation is linear or nonlinear: $$y'' + (y')^3 + y = 0$$. Justify your answer.

Accepted Answer

Nonlinear, due to the presence of the term $(y')^3$.

Answer

Linear, as all terms are of the first order or lower.

Question 18

Classify the following differential equation: y'' + 2y' + 5y = 0

Accepted Answer

Second-order ordinary differential equation (ODE)

Answer

First-order ordinary differential equation (ODE)

Answer

First-order partial differential equation (PDE)

Answer

Second-order partial differential equation (PDE)

Question 19

Identify the integrating factor for the linear ODE y' + P(x)y = Q(x):

Accepted Answer

exp(∫P(x)dx)

Answer

exp(-∫P(x)dx)

Answer

-exp(∫P(x)dx)

Answer

1

Question 20

Which method is appropriate for solving the Cauchy-Euler equation ay'' + by' + cy = 0?

Accepted Answer

Trial solutions

Answer

Method of undetermined coefficients

Answer

Variation of parameters

Answer

Laplace transform

Question 21

Apply the separation of variables technique to solve the heat equation ∂u/∂t = ∂^2u/∂x^2:

Accepted Answer

u(x, t) = X(x)*T(t)

Answer

u(x, t) = X(x) + T(t)

Answer

u(x, t) = X(t)/T(x)

Answer

u(x, t) = X(t)*T(x)

Question 22

Interpret the physical significance of the wave equation ∂^2u/∂t^2 = c^2∂^2u/∂x^2:

Accepted Answer

Propagation of waves

Answer

Propagation of heat

Answer

Diffusion

Answer

Electromagnetic field behavior

Question 23

What distinguishes partial differential equations (PDEs) from ordinary differential equations (ODEs)?

Accepted Answer

PDEs involve partial derivatives with respect to multiple independent variables.

Answer

PDEs are always more complex and difficult to solve than ODEs.

Answer

PDEs have limited applications in science and engineering.

Question 24

Determine the conditions under which the differential equation $$M(x, y)dx + N(x, y)dy = 0$$ is exact. If it is exact, explain how to find its solution.

Accepted Answer

The equation is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. In this case, find $f(x, y)$ such that $\frac{\partial f}{\partial x} = M(x, y)$ and $\frac{\partial f}{\partial y} = N(x, y)$.

Answer

The equation is always exact.

Answer

The solution to an exact differential equation is always $f(x, y) = c$.

Question 25

Which of the following is a first-order linear ordinary differential equation (ODE)?

Accepted Answer

y' + y = x

Answer

y'' + y' + y = 0

Answer

y' = y^2

Answer

y'' = sin(x)

Question 26

Find the general solution to the first-order linear ODE y' - 2y = e^x.

Accepted Answer

y = (1/2)e^x + Ce^(-2x)

Answer

y = e^x + Ce^(-2x)

Answer

y = (1/2)e^(-2x) + Ce^x

Answer

y = e^(-2x) + Ce^x

Question 27

Which of the following is a method for solving exact partial differential equations (PDEs)?

Accepted Answer

Method of characteristics

Answer

Method of separation of variables

Answer

Lagrange multipliers

Answer

Monte Carlo simulation

Question 28

What type of boundary condition specifies the value of the solution at a point?

Accepted Answer

Dirichlet boundary condition

Answer

Neumann boundary condition

Answer

Cauchy boundary condition

Answer

Robin boundary condition

Question 29

Which of the following is a second-order ordinary differential equation (ODE)?

Accepted Answer

y'' = y

Answer

y' + y = sin(x)

Answer

y' = xy

Answer

y''' = x^2

Question 30

Which of the following is a partial derivative of the function u(x, y)?

Accepted Answer

du/dy

Answer

du/dx

Answer

d^2u/dxdy

Answer

d^2u/dydx

Question 31

Which of the following methods can be used to solve the differential equation y' + 2xy = x?

Accepted Answer

Integrating factor

Answer

Separation of variables

Answer

Homogeneous differential equation

Answer

Exact differential equation

Question 32

Determine the type of partial differential equation represented by u_x + u_y + u_z = 0.

Accepted Answer

Elliptic

Answer

Hyperbolic

Answer

Parabolic

Answer

Ordinary

Question 33

Which of the following is the general solution to the differential equation $y' + 2xy = 0$?

Accepted Answer

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

Answer

$y = e^x$

Answer

$y = e^{-2x}$

Answer

$y = x^2$

Question 34

Which of the following is a type of singularity for the differential equation $x^2y'' + xy' + y = 0$?

Accepted Answer

Regular singular point

Answer

Irregular singular point

Answer

Ordinary point

Answer

Essential singularity

Question 35

Find the Frobenius series solution to the differential equation $x^2y'' - xy' + (x^2 - rac{1}{4})y = 0$.

Accepted Answer

Solution involves a series of powers of $x^{1/2}$.

Answer

Solution involves a series of powers of $x$.

Answer

Solution involves a series of powers of $x^2$.

Answer

Solution involves a series of powers of $1/x$.

Question 36

Classify the following differential equation: y'' + 2y' + y = 0

Accepted Answer

Second-order, linear, homogeneous ODE

Answer

First-order, linear, homogeneous ODE

Answer

First-order, non-linear, homogeneous ODE

Answer

Second-order, non-linear, homogeneous ODE

Question 37

Which of the following is a method to solve the partial differential equation ∂u/∂x + ∂u/∂y = u?

Accepted Answer

Method of separation of variables

Answer

Method of characteristics

Answer

Laplace transform

Answer

Fourier transform

Question 38

What is the order of the differential equation: (d^4y/dx^4) - 3(d^2y/dx^2) + 2y = 0?

Accepted Answer

Fourth-order

Answer

First-order

Answer

Second-order

Answer

Third-order

Question 39

Which of the following is NOT an application of differential equations in engineering?

Accepted Answer

Predicting weather patterns

Answer

Modelling electrical circuits

Answer

Analysing heat transfer

Answer

Designing bridges

Question 40

What is the initial condition for the differential equation y' = y, y(1) = 2?

Accepted Answer

y(1) = 2

Answer

y(0) = 1

Answer

y(2) = 4

Answer

y(3) = 8

Question 41

What is the Laplace transform of the function f(t) = t^2?

Accepted Answer

F(s) = 2/(s^3)

Answer

F(s) = 2s/(s^3)

Answer

F(s) = s^2/(s^3)

Answer

F(s) = 2s^2/(s^3)

Question 42

Solve the non-homogeneous differential equation: $$y''-4y'+3y=e^x(x+1)$$ using the method of undetermined coefficients.

Accepted Answer

$y_p=(Ax+B)e^x+Ce^{2x}+De^x$

Answer

$y_p=Ae^x+Be^{2x}+(Cx+D)e^x$

Answer

$y_p=Ae^x+Be^{2x}+Cx^2e^x$

Answer

$y_p=(Ax^2+B)e^x+Ce^{2x}+De^x$

Question 43

In the method of characteristics for solving first-order partial differential equations, what is the significance of the characteristic equations and how do they influence the solution's behavior?

Accepted Answer

They determine the direction of the characteristics and the rate of change of the solution along them.

Answer

They provide the specific values of the constants in the general solution.

Answer

They ensure the obtained solution is continuous and differentiable.

Answer

They allow the solution to be expressed in a closed-form analytical expression.

Question 44

Which of the following is NOT a defining feature of a differential equation?

Accepted Answer

Always has a unique solution

Answer

Involves derivatives of a dependent variable with respect to independent variables

Answer

Involves unknown functions

Answer

May involve ordinary or partial derivatives

Question 45

The order of a differential equation refers to:

Accepted Answer

The highest order derivative present in the equation

Answer

The number of independent variables

Answer

The degree of the polynomial involved

Answer

The number of dependent variables

Question 46

The method of characteristics is commonly used to solve:

Accepted Answer

First-order partial differential equations

Answer

Second-order partial differential equations

Answer

Systems of partial differential equations

Answer

Non-linear partial differential equations

Question 47

The method of separation of variables can be applied to solve which of the following types of partial differential equations?

Accepted Answer

Linear homogeneous partial differential equations with constant coefficients

Answer

Linear inhomogeneous partial differential equations with variable coefficients

Answer

Non-linear homogeneous partial differential equations

Answer

Non-linear inhomogeneous partial differential equations

Question 48

Consider the partial differential equation (PDE) ∂u/∂t + ∂^2u/∂x^2 = 0. What is its order?

Accepted Answer

2

Answer

4

Answer

3

Answer

1

Question 49

Which of the following is a boundary condition type for partial differential equations (PDEs)?

Accepted Answer

Dirichlet boundary condition

Answer

Algebraic equation

Answer

Exponential function

Answer

Trigonometric function

Question 50

Which of the following applications utilizes differential equations in engineering?

Accepted Answer

Modeling heat transfer

Answer

Solving a system of linear equations

Answer

Calculating the area of a circle

Answer

Finding the derivative of a function

Question 51

Which technique is employed to solve non-homogeneous linear ordinary differential equations (ODEs)?

Accepted Answer

Method of undetermined coefficients

Answer

Riccati equation

Answer

Lagrange's equation

Answer

Cauchy-Euler equation

Question 52

Which type of partial differential equation (PDE) describes wave propagation?

Accepted Answer

Wave equation

Answer

Helmholtz equation

Answer

Diffusion equation

Answer

Laplace equation

Question 53

What is the primary purpose of the method of characteristics?

Accepted Answer

Solving first-order partial differential equations (PDEs)

Answer

Solving second-order PDEs

Answer

Solving systems of ordinary differential equations (ODEs)

Answer

Solving integral equations

Question 54

Identify the first-order linear differential equation:

Accepted Answer

y' + 2xy = x

Answer

y'' + 3y' + 2y = sin(x)

Answer

y^2 y' = x^2

Answer

y''' - y' = e^x

Question 55

Identify the second-order homogeneous differential equation with constant coefficients:

Accepted Answer

y'' - 4y' + 3y = 0

Answer

y^2 y' = x^2

Answer

y'' + y' = x

Answer

y''' - y' = e^x

Question 56

Determine the general solution for the differential equation y'' - 2y' + y = e^x:

Accepted Answer

y = C₁e^x + C₂e^(-x) + e^x

Answer

y = C₁e^x - C₂e^(-x) - e^x

Answer

y = C₁e^x + C₂e^(-x) - e^x

Answer

y = C₁e^x - C₂e^(-x) + e^x

Question 57

Identify the boundary value problem for an ordinary differential equation:

Accepted Answer

y'' + y = 0, y(0) = 0, y(1) = 1

Answer

y'' = y

Answer

y' + y = 0

Answer

y''' + y' = e^x

Question 58

Identify an initial value problem for a partial differential equation:

Accepted Answer

∂u/∂t = ∂²u/∂x², u(0, x) = f(x)

Answer

∂u/∂x + ∂u/∂y = 0

Answer

∂²u/∂x² + ∂²u/∂y² = 0

Question 59

Which of the following represents a first-order linear differential equation?

Accepted Answer

y' + x*y = sin(x)

Answer

(y')^2 + y'' = 0

Answer

y'' + 2y' + y = x

Answer

y''' - 3y'' + 2y' - y = x^2

Question 60

The method of undetermined coefficients is primarily used to solve which type of differential equations?

Accepted Answer

Non-homogeneous linear differential equations

Answer

Homogeneous linear differential equations

Answer

Partial differential equations

Answer

Systems of differential equations

Question 61

Which of the following is an example of a characteristic equation for a linear differential equation?

Accepted Answer

r^2 + 2r + 1 = 0

Answer

y'' + y' - 6y = 0

Answer

x^2 - 2x + 1 = 0

Answer

y' - 2y = 0

Question 62

Which of the following is an example of a partial differential equation?

Accepted Answer

u_x + u_y - u = 0

Answer

x*y' = y*x'

Answer

y = c1*e^(2x) + c2*e^(-2x)

Answer

y'' - 2y' + y = 0

Question 63

Which of the following is an example of a boundary condition for a partial differential equation?

Accepted Answer

u(0, t) = sin(t)

Answer

y = c1*e^(2x) + c2*e^(-2x)

Answer

y'' - 2y' + y = 0

Answer

x*y' = y*x'

Question 64

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

Accepted Answer

dy/dx + 2y = x

Answer

dy/dx + y^2 = x

Answer

d^3y/dx^3 + y = 0

Answer

d^2y/dx^2 + y = 0