Master First-Order Differential Equations: A Comprehensive Guide for Electrical Engineering Students

Question 1

Identify the first-order differential equation among the following:

Accepted Answer

dy/dx = y^2

Answer

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

Answer

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

Answer

y'' + y' - y = 0

Question 2

Consider the differential equation: dy/dx + y = f(x). Under which condition is the solution to this equation guaranteed to be unique?

Accepted Answer

f(x) is continuous

Answer

f(x) is differentiable

Answer

f(x) is integrable

Answer

f(x) is bounded

Question 3

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

Accepted Answer

dy/dx = y/x

Answer

dy/dx = (x + y)/(x - y)

Answer

dy/dx = x^2 + y^2

Answer

dy/dx = sin(x*y)

Question 4

For the differential equation y' + P(x)y = Q(x), the integrating factor is given by:

Accepted Answer

e^(∫P(x)dx)

Answer

e^(−∫P(x)dx)

Answer

(∫P(x)dx)^−1

Answer

Q(x)

Question 5

The differential equation y' = xy + y^2 is a:

Accepted Answer

First-order equation

Answer

Second-order equation

Answer

Third-order equation

Answer

Fourth-order equation

Question 6

A linear first-order differential equation has the form:

Accepted Answer

y' + y = cos(x)

Answer

y' = xy + sin(x)

Answer

y' = y^2 + x

Answer

y' + y sin(x) = e^x

Question 7

For the differential equation y' = f(x, y) with f(x, 0) = 0, the solution curves:

Accepted Answer

All pass through the origin (0, 0).

Answer

All are parallel to each other.

Answer

All are perpendicular to each other.

Answer

Form a family of circles centered at the origin.

Question 8

The substitution that can reduce the equation y' = xy^2 + x^2y to a separable equation is:

Accepted Answer

v = y/x

Answer

v = x/y

Answer

v = y^2

Answer

v = x^2

Question 9

In the method of integrating factors used to solve first-order differential equations, the integrating factor is a function of which variable(s)?

Accepted Answer

Both independent and dependent variables (x and y)

Answer

Independent variable (x) only

Answer

Dependent variable (y) only

Answer

Neither x nor y

Question 10

Which condition must be met for a first-order differential equation to be considered exact?

Accepted Answer

The partial derivatives of the function must be continuous.

Answer

The function must be differentiable twice.

Answer

The coefficient of the derivative term must be non-zero.

Question 11

Identify the first-order linear differential equation:

Accepted Answer

y' + y = sin(x)

Answer

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

Answer

y' + x^2y = e^x

Answer

y' = y^2 + x

Question 12

Determine the integrating factor for the exact differential equation: (2x + y)dx + (x + 2y)dy = 0

Accepted Answer

e^(2x+y)

Answer

e^(x+y)

Answer

e^(2x)

Answer

e^(x)

Question 13

Identify the solution to the first-order differential equation: y' + 2y = 5

Accepted Answer

y = (5/2) + Ce^(-2x)

Answer

y = (5/2) - Ce^(-2x)

Answer

y = 5Ce^(-2x)

Answer

y = -Ce^(-2x)

Question 14

Solve the homogeneous differential equation: y' = y/x

Accepted Answer

y = C/x

Answer

y = Cx

Answer

y = x/C

Answer

y = C*ln(x)

Question 15

Identify the integrating factor for the non-exact differential equation: (1 + x^2)dy + 2xy dx = 0

Accepted Answer

1/(1 + x^2)

Answer

e^(x^2)

Answer

e^(2x^2)

Answer

1/x^2

Question 16

Which of the following conditions is necessary for a differential equation Mdx + Ndy = 0 to be exact?

Accepted Answer

The partial derivative of M with respect to y is equal to the partial derivative of N with respect to x.

Answer

The equation is homogeneous.

Answer

The equation is separable.

Answer

The equation is linear.

Question 17

Find the integrating factor for the differential equation (x + y)dy = (x - y)dx.

Accepted Answer

1/(x+y)

Answer

1/(x-y)

Answer

x+y

Answer

x-y

Question 18

Solve the initial value problem: y' + 2xy = 4x, y(0) = 1.

Accepted Answer

y = 2x^2 - 1

Answer

y = x^2 + 1

Answer

y = 2x - 1

Answer

y = x - 1

Question 19

Determine if the differential equation y' = y^2 + x is autonomous.

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Answer

Only when y = 0

Question 20

Consider the equation y' = y/x. Which of the following is true about the equation?

Accepted Answer

It is separable and linear.

Answer

It is exact and homogeneous.

Answer

It is non-homogeneous and Bernoulli.

Answer

It is linear and non-autonomous.

Question 21

Find the solution to the initial value problem y' - 2y = e^x, y(0) = 1

Accepted Answer

y = (e^x - 1)/2

Answer

y = (e^x + 1)/2

Answer

y = e^x/2

Answer

y = -e^x/2

Question 22

Which of the following equations is an exact differential equation?

Accepted Answer

Mdx + Ndy = M_x dx + N_y dy

Answer

Mdx + Ndy = 0

Answer

Mdx - Ndy = 0

Answer

Mdy - Ndx = 0

Question 23

Find the solution to the separable differential equation dy/dx = y/x

Accepted Answer

y = Cx

Answer

y = Cxe^x

Answer

y = Cx^2

Answer

y = C/x

Question 24

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

Accepted Answer

xy' + y = x

Answer

y'' + y' - 2y = 0

Answer

(y')^2 + y^2 = 1

Answer

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

Question 25

Find the integrating factor for the differential equation (x + y)dx + (x - y)dy = 0.

Accepted Answer

xy

Answer

1

Answer

x

Answer

y

Question 26

Which of the following conditions is necessary for a differential equation to be exact?

Accepted Answer

Mdx + Ndy = 0, where M and N have continuous first-order partial derivatives

Answer

Mdx + Ndy = 0, where M and N have continuous second-order partial derivatives

Answer

Mdx - Ndy = 0, where M and N have continuous first-order partial derivatives

Answer

Mdx - Ndy = 0, where M and N have continuous second-order partial derivatives

Question 27

Which of the following equations is a Bernoulli equation?

Accepted Answer

y' + P(x)y = Q(x)y^n

Answer

y'' + P(x)y' + Q(x)y = 0

Answer

(y')^2 + P(x)y' + Q(x)y = 0

Answer

y' + P(x)y = Q(x)y + R(x)

Question 28

Solve the Bernoulli equation: y' - y = xy^2

Accepted Answer

y = 1/(1 - Cx)

Answer

y = 1/(1 + Cx)

Answer

y = C*e^x

Answer

y = C*e^(-x)

Question 29

Which of the following functions is an integrating factor for the differential equation (x^2 + y^2)dx + x y dy = 0?

Accepted Answer

1/(x^2 + y^2)

Answer

1/x

Answer

1/y

Answer

x^2

Question 30

Classify the differential equation y' + p(x)y = q(x):

Accepted Answer

Linear differential equation

Answer

Nonlinear differential equation

Answer

Partial differential equation

Answer

Ordinary differential equation

Question 31

Which technique is commonly employed to solve homogeneous first-order differential equations?

Accepted Answer

Substitution

Answer

Bernoulli's equation

Answer

Exact equations

Answer

Euler's method

Question 32

Which method is NOT used for solving first-order differential equations?

Accepted Answer

Substitution

Answer

Separation of variables

Answer

Integrating factors

Answer

Exact equations

Question 33

Consider the Bernoulli equation y' + P(x)y = Q(x)y^n. Which of the following substitutions can transform it into a linear equation?

Accepted Answer

v = 1/y^(n-1)

Answer

v = y^(n-1)

Answer

v = y

Answer

v = x

Question 34

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

Accepted Answer

y' = 2xy / (x^2 + y^2)

Answer

y' = x/y

Answer

y' = y^2 + x^2

Answer

y' = y + x

Question 35

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

Accepted Answer

Linear Homogeneous Differential Equation

Answer

Exact Differential Equation

Answer

Separable Differential Equation

Answer

Bernoulli Differential Equation

Question 36

For the differential equation dy/dx + Py = Q, which of the following is the integrating factor?

Accepted Answer

e^∫P dx

Answer

e^(-∫P dx)

Answer

∫P dx

Answer

1/P

Question 37

What is the general solution of the differential equation dy/dx = y/x?

Accepted Answer

y = Cx, where C is an arbitrary constant

Answer

y = C/x, where C is an arbitrary constant

Answer

y = x^2 + C, where C is an arbitrary constant

Answer

y = x + C, where C is an arbitrary constant

Question 38

A tank initially contains V0 liters of pure water. Brine containing a fixed concentration of salt enters the tank at a rate of r liters per minute. The well-stirred mixture leaves the tank at the same rate. Which of the following differential equations models the amount of salt, S(t), in the tank at time t?

Accepted Answer

dS/dt = rCin - (r/V)S, where Cin is the salt concentration in the incoming brine

Answer

dS/dt = (r/V)S + rCin, where Cin is the salt concentration in the incoming brine

Answer

dS/dt = (r/V)S - rCin, where Cin is the salt concentration in the incoming brine

Answer

dS/dt = rCin + (r/V)S, where Cin is the salt concentration in the incoming brine

Question 39

The initial value problem y' = y, y(0) = 2 has the following solution:

Accepted Answer

y = 2e^x

Answer

y = x^2 + 2

Answer

y = x + 2

Answer

y = 2x - 1

Question 40

What is the order of the differential equation y''' + 2y'' - 5y' + 6y = 0?

Accepted Answer

3

Answer

4

Answer

2

Answer

1

Question 41

The differential equation dy/dx = f(x, y) is said to be autonomous if:

Accepted Answer

f(x, y) does not depend explicitly on x

Answer

f(x, y) is a linear function of x and y

Answer

f(x, y) is a constant function

Question 42

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

Accepted Answer

It can be written in the form dy/dx + P(x)y = Q(x), where P(x) and Q(x) are continuous functions.

Answer

It involves only first derivatives of the dependent variable.

Answer

It has a constant solution.

Answer

It is an equation of degree one in the dependent variable.

Question 43

Consider the circuit represented by the differential equation L(dI/dt) + RI = V. What is the order of this equation?

Accepted Answer

First order

Answer

Third order

Answer

Zero order

Answer

Second order

Question 44

Find the solution to the differential equation dy/dx + y/x = x^2.

Accepted Answer

y = (x^3/4) + (C/x)

Answer

y = (x^3/4) - (C/x)

Answer

y = (x^3/4) + Cx

Answer

y = (x^3/4) + C

Question 45

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

Accepted Answer

dy/dx + 2y = x

Answer

dy/dx = y² + x

Answer

d²y/dx² + y = sin(x)

Answer

dy/dx + y² = 0

Question 46

The general solution of the differential equation dy/dx = x/y is:

Accepted Answer

y² = x² + C

Answer

y² = 2x + C

Answer

y = x + C

Answer

y = x² + C

Question 47

The integrating factor for the differential equation dy/dx + 2y = e^x is:

Accepted Answer

e^(2x)

Answer

e^(-x)

Answer

e^(-2x)

Answer

e^x

Question 48

A first-order differential equation is called exact if:

Accepted Answer

∂M/∂y = ∂N/∂x

Answer

∂M/∂x + ∂N/∂y = 0

Answer

∂M/∂x = ∂N/∂y

Answer

∂M/∂y - ∂N/∂x = 0

Question 49

The solution of the initial value problem dy/dx = y^2, y(0) = 1 is:

Accepted Answer

y = 1/(1-x)

Answer

y = 1 + x

Answer

y = 1/(1+x)

Answer

y = 1 - x

Question 50

A tank initially contains 100 liters of pure water. Brine containing 2 kg of salt per liter enters the tank at a rate of 5 liters per minute. The well-mixed solution leaves the tank at the same rate. What is the differential equation that models the amount of salt in the tank at any time t?

Accepted Answer

dS/dt = 10 - S/20

Answer

dS/dt = 10 + S/20

Answer

dS/dt = 5 + S/20

Answer

dS/dt = 5 - S/20

Question 51

Which of the following is a possible application of first-order differential equations?

Accepted Answer

Modeling population growth

Answer

Finding the roots of a polynomial

Answer

Solving a system of linear equations

Answer

Calculating the area of a triangle

Question 52

The direction field of a first-order differential equation represents:

Accepted Answer

The slope of the solution curves at various points in the plane

Answer

The integral curves of the differential equation

Answer

The general solution of the differential equation

Question 53

A first-order differential equation is said to be autonomous if:

Accepted Answer

The independent variable does not appear explicitly in the equation

Answer

The dependent variable does not appear explicitly in the equation

Answer

The equation is linear

Question 54

The equilibrium solutions of an autonomous first-order differential equation are found by:

Accepted Answer

Setting the right-hand side of the equation equal to zero

Answer

Solving the differential equation

Answer

Finding the critical points of the solution