Test Your Skills

Which equation represents the separable form of the given differential equation?

$(P(1, v) + vQ(1, v)) dx + xQ(1, v) dv = 0$
$(x^2 + x^2v^2) dx + x^2v(v dx + x dv) = 0$
$P(x, y) dx + Q(x, y) dy = 0$
$(1 + 2v^2) dx + xv dv = 0$ (correct)

Which substitution is made to find the solution for any homogeneous equation?

$y = xv$ (correct)
$v = y/x$
$v = x/y$
$y = vx$

Which term is used to transform the given equation into a separable form?

Integrating factor (correct)
Collecting terms
Homogeneity
Common factor

Which equation represents the final answer after substituting $v = y/x$?

$x^4 + 2x^2 y^2 = C$ (C) Signup and view all the answers

What is the integrating factor for the given differential equation?

$\frac{1}{x(1 + 2v^2)}$ (B) Signup and view all the answers

Which type of differential equations are explored in this section?

Differential forms (A) Signup and view all the answers

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 (B) Signup and view all the answers

What is the language used to describe the symmetrized differential equation?

Differential forms (A) Signup and view all the answers

What is the differential form in the two variables $x$ and $y$?

$\omega = P(x, y) , dx + Q(x, y) , dy$ (D) Signup and view all the answers

When is $y$ a solution to the differential equation $P(x, y) + Q(x, y) \frac{dy},{dx} = 0$?

When $P(x, y) + Q(x, y) \frac{dy},{dx} = 0$ (A) Signup and view all the answers

Which equation represents the differential form variant of the differential equation $\omega = x dx + y dy = 0$?

$\frac{dy},{dx} = -\frac{x},{y}$ (D) Signup and view all the answers

What is the equation that represents the solutions defined implicitly by the differential equation $\omega = x dx + y dy = 0$?

$x^2 + y^2 = C$ (B) Signup and view all the answers

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 (C) Signup and view all the answers

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 (D) Signup and view all the answers

What is the differential form called if it is the differential of a continuously differentiable function?

Exact differential form (B) Signup and view all the answers

Which of the following is a criterion for exactness of a differential equation?

$\partial \partial y (\mu P) = \partial \partial x (\mu Q)$ (A) Signup and view all the answers

Which of the following statements is true about the integrating factor for a separable equation?

It depends only on x. (B) Signup and view all the answers

What is the integrating factor for the equation $(xy - 2)dx + (x^2 - xy)dy = 0$?

$\mu = 1/x$ (A) Signup and view all the answers

What is the general solution to the equation $(xy - 2)dx + (x^2 - xy)dy = 0$?

$F(x, y) = xy - 2 \ln|x| + \phi(y)$ (A) Signup and view all the answers

When is a differential equation said to be homogeneous?

When both coefficients are homogeneous of the same degree. (D) Signup and view all the answers

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

$1/x^3$ (D) Signup and view all the answers

Which type of differential equation has the form $dy/dx = a(x)y + f(x)$?

Linear equation (A) Signup and view all the answers

What is the general solution to the linear equation $[a(x)y + f(x)]dx - dy = 0$?

$y(x) = \int e^{-a(x)}f(x)dx$ (C) Signup and view all the answers

What is the solution to the separable equation $-y^2dx + x^3dy = 0$?

$y(x) = \frac{2x^2},{1 - 2Cx^2}$ (C) Signup and view all the answers

What is the condition for a differential equation to be separable?

The equation must have separated variables (D) Signup and view all the answers

Which equation represents a differential form that is exact?

$x^2,dx + y^2,dy$ (D) Signup and view all the answers

What is the condition for a differential form to be exact?

$\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}$ (D) Signup and view all the answers

What is the solution to the differential equation $P(x, y)dx + Q(x, y)dy = 0$ when it is exact?

$F(x, y) = C$ (B) Signup and view all the answers

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?