Explanation

In the power set P(X), the intersection operation is defined as:

A ∩ B = {x ∈ X | x ∈ A and x ∈ B}

The identity element with respect to intersection is the element that does not change the result when intersected with any other element. In this case, the identity element is the set X itself, because:

A ∩ X = A

for any subset A of X.

Explanation

The correct answer is: ∑1/n^p converges if p > 1.

The p-series ∑1/n^p is a well-known series in mathematics, and its convergence properties are as follows:

• If p > 1, the series converges.
• If p ≤ 1, the series diverges.

This is because the terms of the series decrease slower than the terms of a geometric series with ratio less than 1 when p ≤ 1, and decrease faster than the terms of a geometric series with ratio less than 1 when p > 1.

Explanation

• The set S = {1/n : n ∈ N} is a set of positive fractions, where n is a natural number.
• As n increases, the value of 1/n decreases and approaches 0.

Explanation

The given equation y = Asin(x) + Bcos(x) is a general solution to the differential equation y'' + y = 0, where A and B are arbitrary constants.

Explanation

The given differential equation is: x²dy + y²dx = 0

This is a first-order differential equation because it involves only the first derivative of y (i.e., dy) and not higher-order derivatives (such as d²y/dx²).

To see this more clearly, we can rewrite the equation as: dy/dx = -y²/x², which is a first-order differential equation in the form dy/dx = f(x,y).

Explanation

The Differential Equation ydx-2xdy=0 represents the family of 'Parabola'.

Explanation

The given differential equation is a second-order linear homogeneous equation with constant coefficients. The characteristic equation is:

r^2 - 4r + 4 = 0

which has a repeated root r = 2. Therefore, the general solution is:

y = (c1 + c2x)e^2x

where c1 and c2 are arbitrary constants.

Explanation

A Clairaut equation is a type of differential equation that can be solved by substituting dy/dx = p, which reduces the equation to a linear differential equation in x and y.

Explanation

The given differential form can be simplified as: xdy + ydx/xy = d(ln(xy))

Here's a step-by-step solution:

xdy + ydx/xy = (x/y)dy + (y/x)dx

= d(ln(y)) + d(ln(x))

= d(ln(xy))

Explanation

A Bernoulli equation is a nonlinear differential equation that can be transformed into a linear equation by substituting v = y^(1-n), where n is the coefficient of the y² term (in this case, n = 2).