The characteristic equation is r² - 3r + 2 = 0. Solving for r gives roots r = 1 and r = 2. Hence, the general solution is y(x) = C₁e^x + C₂e^(2x).

Using spherical coordinates, the integral simplifies to ∭_V (ρ²)ρ²sin(θ) dρ dθ dφ, and evaluating gives the result (4π/5).

The characteristic equation is (λ - 2)² - 1 = 0, which gives eigenvalues λ = 3 and λ = 1. The corresponding eigenvectors are v₁ = [1, 1] and v₂ = [-1, 1].

Using the p-series test with p = 2 (where p > 1), the series ∑ (1/n²) converges.

The general solution is u(x, y) = F(x + iy) + G(x - iy), where F and G are arbitrary analytic functions.

The Fourier series of f(x) = x² is given by a₀ + ∑ (aₙcos(nx)), where a₀ and aₙ are the Fourier coefficients. The calculation results in a specific Fourier expansion.

Using the residue theorem, the residue at z = 1 is 2, so the contour integral evaluates to 4πi.

Since the determinant of a skew-symmetric matrix equals its negative and the matrix has odd order, the determinant must be zero.

The Taylor series of f(x) = e^x at x = 0 is ∑ (xⁿ/n!), where the sum runs from n = 0 to infinity.

The integral converges and is known as the Gaussian integral. Its value is √π/2.