Question 1

Consider the differential equation $\frac{dy}{dx} + y = x$. Which of the following is the integrating factor for this equation?

$e^{-x}$
$e^x$
$x$
$1$

Answer

Correct Option: B

Question 2

Solve the following exact equation: $\left(2x + 3y\right)dx + \left(3x - 2y\right)dy = 0$.

$x^2 + 3xy - y^2 = C$
$x^2 - 3xy + y^2 = C$
$2x^2 + 3xy - 2y^2 = C$
$2x^2 - 3xy + 2y^2 = C$

Answer

Correct Option: A

Question 3

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

$y' + y = x^2$
$y'' + 2y' + y = 0$
$y' = x^2 + y^2$
$y''' - 3y'' + 2y' - y = 0$

Answer

Correct Option: A

Question 4

Use the method of separation of variables to solve the following differential equation: $\frac{dy}{dx} = \frac{x}{y}$.

$y^2 = x^2 + C$
$y = x^2 + C$
$y = \ln(x) + C$
$y = x + C$

Answer

Correct Option: A

Question 5

Consider the differential equation $\frac{dy}{dx} = \frac{y}{x} + \frac{x}{y}$. Which of the following is a suitable substitution to solve this equation?

$y = vx$
$y = x^v$
$y = \ln(x)$
$y = e^x$

Answer

Correct Option: A

Question 6

Find the general solution of the differential equation $\frac{dy}{dx} = \frac{2x + y}{x}$.

$y = x^2 + C$
$y = x + C$
$y = \ln(x) + C$
$y = \frac{1}{x} + C$

Answer

Correct Option: A

Question 7

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

$y' + y = x$
$y' = xy$
$y'' + 2y' + y = 0$
$y' = \frac{y}{x}$

Answer

Correct Option: D

Question 8

Solve the following homogeneous differential equation: $\frac{dy}{dx} = \frac{y - x}{y + x}$.

$y = x + C$
$y = x^2 + C$
$y = \ln(x) + C$
$y = \frac{1}{x} + C$

Answer

Correct Option: A

Question 9

Consider the differential equation $\frac{dy}{dx} = \frac{x^2 + y^2}{xy}$. Which of the following is a suitable substitution to solve this equation?

$y = vx$
$y = x^v$
$y = \ln(x)$
$y = e^x$

Answer

Correct Option: A

Question 10

Find the general solution of the differential equation $\frac{dy}{dx} = \frac{2x + 3y}{x + 2y}$.

$y = x + C$
$y = x^2 + C$
$y = \ln(x) + C$
$y = \frac{1}{x} + C$

Answer

Correct Option: A

Question 11

Which of the following is a Bernoulli differential equation?

$y' + y = x$
$y' = xy$
$y'' + 2y' + y = 0$
$y' = \frac{y}{x} + \frac{x}{y}$

Answer

Correct Option:

Question 12

Solve the following Bernoulli differential equation: $\frac{dy}{dx} = x^2y - y^2$.

$y = \frac{1}{x} + C$
$y = \frac{1}{x^2} + C$
$y = \ln(x) + C$
$y = \frac{1}{x^3} + C$

Answer

Correct Option: A

Question 13

Consider the differential equation $\frac{dy}{dx} = \frac{y^2 + x^2}{xy}$. Which of the following is a suitable substitution to solve this equation?

$y = vx$
$y = x^v$
$y = \ln(x)$
$y = e^x$

Answer

Correct Option: A

Question 14

Find the general solution of the differential equation $\frac{dy}{dx} = \frac{x + y}{x - y}$.

$y = x + C$
$y = x^2 + C$
$y = \ln(x) + C$
$y = \frac{1}{x} + C$

Answer

Correct Option: A

Question 15

Which of the following is a Riccati differential equation?

$y' + y = x$
$y' = xy$
$y'' + 2y' + y = 0$
$y' = \frac{y}{x} + \frac{x}{y}$

Answer

Correct Option:

Description: This quiz covers the fundamental concepts and techniques related to First Order Differential Equations. It aims to assess your understanding of various methods for solving these equations, including exact equations, integrating factors, and separation of variables.
Number of Questions: 15
Created by: Aliensbrain Bot
Tags: first order differential equations exact equations integrating factors separation of variables

First Order Differential Equations

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