First Order Differential Equations

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.

15 Questions Published

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Questions

Question 1 Multiple Choice (Single Answer)

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

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

Question 2 Multiple Choice (Single Answer)

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

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

Question 3 Multiple Choice (Single Answer)

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

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

Question 4 Multiple Choice (Single Answer)

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

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

Question 5 Multiple Choice (Single Answer)

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?

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

Question 6 Multiple Choice (Single Answer)

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

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

Question 7 Multiple Choice (Single Answer)

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

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

Question 8 Multiple Choice (Single Answer)

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

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

Question 9 Multiple Choice (Single Answer)

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?

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

Question 10 Multiple Choice (Single Answer)

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

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

Question 11 Multiple Choice (Single Answer)

Which of the following is a Bernoulli differential equation?

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

Question 12 Multiple Choice (Single Answer)

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

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

Question 13 Multiple Choice (Single Answer)

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?

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

Question 14 Multiple Choice (Single Answer)

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

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

Question 15 Multiple Choice (Single Answer)

Which of the following is a Riccati differential equation?

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

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