30 Questions
Which equation represents the separable form of the given differential equation?
$(1 + 2v^2) dx + xv dv = 0$
Which substitution is made to find the solution for any homogeneous equation?
$y = xv$
Which term is used to transform the given equation into a separable form?
Integrating factor
Which equation represents the final answer after substituting $v = y/x$?
$x^4 + 2x^2 y^2 = C$
What is the integrating factor for the given differential equation?
$\frac{1}{x(1 + 2v^2)}$
Which type of differential equations are explored in this section?
Differential forms
What is the purpose of allowing solutions defined implicitly by equations of the form $F(x, y) = C$?
To treat the variables $x$ and $y$ symmetrically
What is the language used to describe the symmetrized differential equation?
Differential forms
What is the differential form in the two variables $x$ and $y$?
$\omega = P(x, y) , dx + Q(x, y) , dy$
When is $y$ a solution to the differential equation $P(x, y) + Q(x, y) \frac{dy},{dx} = 0$?
When $P(x, y) + Q(x, y) \frac{dy},{dx} = 0$
Which equation represents the differential form variant of the differential equation $\omega = x dx + y dy = 0$?
$\frac{dy},{dx} = -\frac{x},{y}$
What is the equation that represents the solutions defined implicitly by the differential equation $\omega = x dx + y dy = 0$?
$x^2 + y^2 = C$
What are the solution curves of the differential equation $\omega = x dx + y dy = 0$?
The level set defined by $x^2 + y^2 = C$ contains two solution curves
What are the integral curves of a differential equation defined by the equation $F(x, y) = C$?
The level sets defined by $F(x, y) = C$ are called integral curves of the differential equation
What is the differential form called if it is the differential of a continuously differentiable function?
Exact differential form
Which of the following is a criterion for exactness of a differential equation?
$\partial \partial y (\mu P) = \partial \partial x (\mu Q)$
Which of the following statements is true about the integrating factor for a separable equation?
It depends only on x.
What is the integrating factor for the equation $(xy - 2)dx + (x^2 - xy)dy = 0$?
$\mu = 1/x$
What is the general solution to the equation $(xy - 2)dx + (x^2 - xy)dy = 0$?
$F(x, y) = xy - 2 \ln|x| + \phi(y)$
When is a differential equation said to be homogeneous?
When both coefficients are homogeneous of the same degree.
Which of the following is an integrating factor for the differential equation $(x + 2y^2)dx - 2xydy = 0$?
$1/x^3$
Which type of differential equation has the form $dy/dx = a(x)y + f(x)$?
Linear equation
What is the general solution to the linear equation $[a(x)y + f(x)]dx - dy = 0$?
$y(x) = \int e^{-a(x)}f(x)dx$
What is the solution to the separable equation $-y^2dx + x^3dy = 0$?
$y(x) = \frac{2x^2},{1 - 2Cx^2}$
What is the condition for a differential equation to be separable?
The equation must have separated variables
Which equation represents a differential form that is exact?
$x^2,dx + y^2,dy$
What is the condition for a differential form to be exact?
$\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}$
What is the solution to the differential equation $P(x, y)dx + Q(x, y)dy = 0$ when it is exact?
$F(x, y) = C$
What is the solution to the differential equation $P(x, y)dx + Q(x, y)dy = 0$ when it is exact, using the alternate method?
$F(x, y) = C$
What is the condition for a differential form to be exact in a rectangle R?
$\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}$
Quiz: Exact Differential Equations and Integrating Factors Test your knowledge of exact differential equations and integrating factors with this quiz. Learn how to solve differential equations using the method of finding functions μ and F that satisfy ∂F ∂x = μP and ∂F ∂y = μQ. Sharpen your skills in solving equations and understanding the concept of integrating factors.
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