Explanation

The given differential form can be rewritten as:

xdy - ydx/x^2 = d(y/x) - (y/x^2)dx

Using the quotient rule for derivatives, we can simplify this to:

d(y/x) - (y/x^2)dx = d(ln(y/x))

Explanation

The integrating factor for a first-order linear differential equation dy/dx + P(x)y = Q(x) is exp (∫P(x)dx).

This is a well-known result in differential equations.

Explanation

If f(x,y) is a homogeneous function of degree zero in x and y, then it satisfies the property: f(ax,ay) = f(x,y)

for any non-zero constant a.

Since the degree is zero, we can write:

f(x,y) = f(1, y/x)

This shows that f(x,y) is a function of y/x alone.

Explanation

The given differential equation is:

dy/dx = 1/√(1+x^2)

To solve this equation, we can use the substitution u = 1+x^2, which implies du/dx = 2x. Then, we can rewrite the equation as:

dy/dx = 1/√u

dy = 1/√u du/2x

2y = ∫1/√u du

Now, we can recognize the integral as the inverse hyperbolic sine (arsinh or sinh^-1) function:

2y = sinh^-1(u) + C

2y = sinh^-1(1+x^2) + C

Dividing both sides by 2, we get:

y = (1/2)sinh^-1(1+x^2) + C/2

Since C/2 is also a constant, we can write the solution as:

y = (1/2)sinh^-1(1+x^2) + c

or simply: y = sinh^-1(x) + c

Explanation

• A subset F of a metric space X is closed if and only if it is equal to its closure (F = F°).
• This means that F contains all its limit points, and no other points.

Explanation

The set of limit points of the open interval ]0,1[ in the real line R is the closed interval [0,1].

This is because every point in [0,1] is a limit point of ]0,1[, as every neighborhood of each point in [0,1] contains a point in ]0,1[.

Explanation

Ext(A) represents the exterior of set A, which is the set of all points in the universal set X that are not in the closure of A (denoted by A°).

The interior of the closure of A (Int(A°)) consists of all points in A° that are not boundary points.

Since the exterior of A consists of all points in X that are not in A°, Ext(A) is equal to Int(A°).

Explanation

• An open ball in the real line R is an open interval, denoted by (a, b) = {x | a < x < b}, where a and b are real numbers and a < b.
• This means that the endpoints a and b are not included in the interval.

Explanation

The notation 7 ∈ Sr (5) indicates that point 7 belongs to the circle Sr (5), which means it is on the circle.

Since point 7 is on the circle, the distance between the center (point 5) and point 7 is equal to the radius r.

The correct answer is d(5,7) < r.

Explanation

The formula p ∨ (p ∧ q) can be simplified using the distributive property of ∨ over ∧, which states that p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r).

Applying this property, we get:

p ∨ (p ∧ q) = (p ∨ p) ∧ (p ∨ q) = p ∧ (p ∨ q) = p