Master First-Order Differential Equations in Engineering Mathematics

Question 1

Select the method used to solve the first-order separable differential equation y' + xy = 0:

Accepted Answer

Separation of variables

Answer

Integrating factor

Answer

Homogeneous equations

Answer

Variation of parameters

Question 2

For the differential equation y' + y/x = e^x, the corresponding integrating factor is:

Accepted Answer

e^x

Answer

x

Answer

1/x

Answer

1/e^x

Question 3

Which of the following differential equations can be solved using the separation of variables method?

Accepted Answer

xy' = y - x

Answer

y' + xy = sin(x)

Answer

y' = x^2 + y^2

Answer

y' = e^(x+y)

Question 4

Determine the solution to the initial value problem y' + y/x = x^2, y(1) = 2.

Accepted Answer

y = x^3 - x^2 + 1

Answer

y = x^3 + x^2 + 1

Answer

y = x^3 - x + 1

Answer

Cannot be solved with the given initial condition

Question 5

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

Accepted Answer

Second-order linear equations

Answer

Separable equations

Answer

Exact equations

Answer

Integrating factor equations

Question 6

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

Accepted Answer

y = (x+1)e^x

Answer

y = (x-1)e^x

Answer

y = (x+1)e^(-x)

Answer

y = (x-1)e^(-x)

Question 7

Which technique is used to solve a Bernoulli equation?

Accepted Answer

Integrating factors

Answer

Separation of variables

Answer

Substitution

Answer

Exact equations

Question 8

Solve the differential equation y' = y^2 and state its type:

Accepted Answer

Separable, with solution y = 1/(x+C)

Answer

Exact, with solution y = e^(x+C)

Answer

Linear, with solution y = Ce^(-x)

Answer

Bernoulli, with solution y = (1+Cx)^(-1/C)

Question 9

Classify the differential equation (1 + y^2)dx + (x - y)dy = 0:

Accepted Answer

Exact

Answer

Linear

Answer

Separable

Answer

Bernoulli

Question 10

Solve the following separable equation:

$y' = (x + y)(x - 2)$

Accepted Answer

y(x) = -2 + Ce^{x²/2 - 2x}

Answer

y(x) = 2 + Ce^{x²/2 + 2x}

Answer

y(x) = -2 + Ce^{-x²/2 + 2x}

Question 11

Consider the following differential equation:

$y' - 2y = xe<sup>2x}$

What type of equation is this?

Accepted Answer

First-order linear equation

Answer

Second-order linear equation

Answer

Nonlinear equation

Answer

Homogeneous equation

Question 12

Which of the following is a method for solving a first-order differential equation?

Accepted Answer

Integrating factor method

Answer

Laplace transform method

Answer

Eigenvalue method

Question 13

What is the order of the following differential equation:

$$rac{d^3y}{dx^3} + rac{d^2y}{dx^2} - 2rac{dy}{dx} + y = 0$$

Accepted Answer

Third order

Answer

Fourth order

Answer

First order

Answer

Second order

Question 14

Which of the following differential equations is of the form y' = f(x,y)?

Accepted Answer

y' = x²y - y

Answer

y'' = x² + y²

Answer

y' = x² + y²

Question 15

Solve the separable differential equation: dy/dx = 2x * y

Accepted Answer

y = Ce^(x^2)

Answer

y = C / e^(x^2)

Answer

y = C * e^(2x)

Answer

y = Ce^(-x^2)

Question 16

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

Accepted Answer

e^(2x)

Answer

e^(-2x)

Answer

x + y

Answer

1 / e^(2x)

Question 17

Solve the initial value problem: dy/dx = x * y, y(0) = 1

Accepted Answer

y = e^(x^2 / 2)

Answer

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

Answer

y = e^(-x^2 / 2)

Answer

y = Ce^(x^2 / 2)

Question 18

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

Accepted Answer

dy/dx + p(x) * y = q(x)

Answer

(dy/dx)^2 + p(x) * y = q(x)

Answer

d^2y/dx^2 + p(x) * dy/dx + q(x) * y = 0

Answer

dy/dx = p(x) * y + q(x)

Question 19

Which of the following is the standard form of a first-order, linear differential equation?

Accepted Answer

y' + p(x)y = q(x)

Answer

y''' + p(x)y'' + q(x)y' + r(x)y = 0

Answer

y'' + p(x)y' + q(x)y = 0

Answer

y'y'' + p(x)y' + q(x)y = 0

Question 20

When solving separable differential equations, which method is commonly employed?

Accepted Answer

Separation of variables

Answer

Integrating factor

Answer

Variation of parameters

Answer

Exact equation

Question 21

In the context of exact differential equations, what does the integrating factor technique involve?

Accepted Answer

Multiplying the equation by a factor that makes it exact.

Answer

Integrating the equation twice to obtain the general solution.

Answer

Converting the equation into a separable differential equation.

Answer

Substituting the equation with a linear differential equation.

Question 22

Which of the following differential equations falls under the category of Bernoulli differential equations?

Accepted Answer

y' + p(x)y = q(x)y^n

Answer

y' + p(x)y = q(x)y

Answer

y'y'' + p(x)y' + q(x)y = 0

Answer

y'' + p(x)y' + q(x)y = 0

Question 23

For the differential equation (2x + 3y)dx + (x + 2y)dy = 0, what is the integrating factor?

Accepted Answer

1/(x + y)

Answer

(x + y)

Answer

1/y

Answer

1/x