Exact Differential Equations: Conditions and Solutions

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What is the key characteristic of a differential equation?

Must be a linear equation

Contains integrals

Involves derivatives, dependent variables, and independent variables (correct)

Only involves independent variables

What type of differential equation is dy + P(x)y = Q(x)?

Linear differential equation (correct)

Nonlinear differential equation

Quasi-linear differential equation

Partial differential equation

What is the general solution in the context of differential equations?

A solution with no arbitrary constants

A solution without dependent variables

A solution that contains arbitrary constants (correct)

A solution that ignores integration

How can Bernoulli’s equation be transformed into a linear differential equation?

By dividing throughout by y^n Signup and view all the answers

What is the solution of y^(1-n) dy + P(y^(1-n)) = Q?

y^(1-n) * Q + C Signup and view all the answers

In the context of ordinary differential equations, what does IF represent in the solution formula y(IF) = ∫(IF)Q + c?

Integration factor Signup and view all the answers

In the context of orthogonal trajectories, what does the function F(y) represent?

The integrating factor Signup and view all the answers

For orthogonal trajectories of electric lines of force between opposite charges, what is the characteristic shape of equipotential lines?

Circles Signup and view all the answers

What is the role of constant 'e' in finding orthogonal trajectories?

Ensuring orthogonality to each other Signup and view all the answers

How are equipotential lines related to streamlines in fluid dynamics?

Perpendicular Signup and view all the answers

In the context of orthogonal trajectories, what characteristic does the family of confocal conics exhibit?

Elliptical form Signup and view all the answers

For orthogonal trajectories of electrical lines between two concentric cylinders, what shape do the equipotential lines acquire?

Rectangles Signup and view all the answers

For the differential equation $y^2y' - y^3\tan(x) - \sin(x)\cos(2x) = 0$, what is the order of the differential equation?

First order Signup and view all the answers

In the context of the flu virus spreading in a college, what does it mean for a differential equation to be exact?

Its solution is in the form of a function Signup and view all the answers

In the context of the flu virus spreading, what does 'Bernoulli’s equation' refer to?

A type of nonlinear first-order differential equation Signup and view all the answers

What is the initial condition given for the differential equation $2xy^3y' + 3(y\log(x) - 2)y - x,dy = 0$?

$y(1) = 2$ Signup and view all the answers

In the differential equation $dy = \frac{x^2 + xy}{x + xy}dx$, what method can be used to solve it?

Separation of variables Signup and view all the answers

What is the necessary and sufficient condition for the differential equation Mdx + Ndy = 0 to be exact?

∂M/∂N = 0 Signup and view all the answers

In the context of exact differential equations, what does the term ∂y/∂x represent?

Partial derivative of y with respect to x Signup and view all the answers

Which of the following is a method to solve exact differential equations?

Separation of variables Signup and view all the answers

What is the solution method provided for the differential equation x^4 - 2xy^2 + y^4 dx - (2x^2 y - 4xy^3 + sin(y)) dy = 0?

Integrate terms of M with respect to x and terms of N with respect to y Signup and view all the answers

In the context of differential equations, what does 'reducible to the exact form' refer to?

'Exact form' after simplification Signup and view all the answers

For a differential equation that is reducible to the exact form, what does ∫f(x)dx represent?

'f(x)' as a constant term Signup and view all the answers

In a body where the isotherms are constant, what do the orthogonal trajectories represent?

Curves along which heat will be flowing in regions free of heat sources or sinks and filled with homogeneous material Signup and view all the answers

For isotherms of a lamina in the xy-plane defined by a certain equation, what do their orthogonal trajectories represent?

Flux lines Signup and view all the answers

Which point lies on both the isotherm and the flux line passing through it?

(-1,1) Signup and view all the answers

How can the family of curves be described if they are self-orthogonal?

Confocal conics Signup and view all the answers

Which of the following equations represents the orthogonal trajectory of the curve $r = 2a(1 + \cos heta)$?

$r = 2a(1 - \cos heta)$ Signup and view all the answers

What parameter characterizes the family of curves that are self-orthogonal?

Parameter of the confocal conics Signup and view all the answers