The equation x^2 = 12y can be rewritten as (y - 0) = (1/3)(x - 0)^2, indicating the focus is at (0, 3).

The standard form of a parabola is y^2 = 4px. Here, 4p = 12, so p = 3. The focus is at (p, 0) = (3, 0).

This is a separable equation. Integrating gives ln|y| = 2x + C, hence y = Ce^(2x).

This is a separable equation. Integrating gives ln|y| = x + C, hence y = Ce^x.

The characteristic equation is r^2 - 5r + 6 = 0, giving roots 2 and 3. Thus, y = C1 e^(2x) + C2 e^(3x).

The general solution is x = (2n+1)π/4, where n is any integer.

Factoring gives sin(x)(1 + 2cos(x)) = 0, leading to x = nπ or cos(x) = -1/2.

The general solution is x = (2n+1)π/3, where n is an integer.

The general solution is x = (2n+1)π/3, where n is an integer.

The general solutions are x = 7π/6 + 2nπ and x = 11π/6 + 2nπ.

Using the identity sin(a) = sin(b) gives x = nπ or x = (2n+1)π/3.

The general solution is x = nπ + (-1)^n π/4, where n is any integer.

This is a first-order linear differential equation. The integrating factor is e^(-3x).

The characteristic equation is r^2 - 5r + 6 = 0, giving roots 2 and 3. Thus, y = C1 e^(2x) + C2 e^(3x).

The integral ∫(2x + 3)dx = x^2 + 3x + C.

The integral ∫(2x^3 - 4x + 1)dx = (1/2)x^4 - 2x^2 + x + C.

The integral of 1/x is ln|x| + C.

Using partial fraction decomposition, we can integrate to find that ∫ (2x + 1)/(x^2 + x) dx = ln|x^2 + x| + C.

Using the identity tan^2(x) = sec^2(x) - 1, we find that ∫ (tan(x))^2 dx = tan(x) - x + C.

The integrand simplifies to x + 1. Therefore, ∫ (x + 1) dx = (1/2)x^2 + x + C.

The integral of sin(kx) is -1/k * cos(kx). Here, k = 2, so ∫ sin(2x) dx = -cos(2x)/2 + C.

f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3). Critical points are x = 0 and x = 3. Test intervals: f' is positive in (0, 3) and (3, ∞).

The inverse of A is (1/det(A)) * adj(A) = (1/-2) * [[4, -2], [-3, 1]] = [[-2, 1]; [1.5, -0.5]].

The length of the latus rectum of a parabola y^2 = 4px is given by 4p. Here, 4p = 16, so p = 4. Therefore, the length of the latus rectum is 4 * 4 = 16.

Length = √[(2 - (-1))² + (3 - (-1))²] = √[3² + 4²] = √25 = 5.

Length = √[(4-1)² + (5-1)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.

Length = |5 - 2| = 3.

Using the limit property, lim x->0 (sin(kx)/x) = k. Here, k = 3, so the limit is 3.

Using L'Hôpital's rule, the limit evaluates to 3.

Using L'Hôpital's rule, lim(x→0) (sin(5x)/x) = 5.