Answer: This response provides detailed solutions to three distinct ordinary differential equation problems, covering various techniques commonly encountered in mathematical methods for physics. Sub-question (i) involves solving an exact differential equation by recognizing a total differential. Sub-question (ii) addresses a second-order linear non-homogeneous differential equation with constant coefficients, utilizing the method of undetermined coefficients for both non-resonant and resonant forcing te...

Answer: The motion of the parachutist is governed by Newton's Second Law, F_net = Ma, as discussed in PHE-05 for systems with resistive forces. The forces acting on the parachutist are gravity, acting downwards, and air resistance, acting upwards and opposing the motion. Taking the downward direction as positive, the net force is the difference between gravitational force and air resistance. Given the mass of the man and parachute as M, the gravitational force is Mg. The air resistance is given as (Mv²...

Answer: The oscillation of a cylinder floating vertically in water, when slightly depressed and released, is a classic example of simple harmonic motion (SHM). This motion occurs because the change in submerged volume due to displacement creates a restoring buoyant force. This force acts to return the cylinder to its equilibrium position. When the cylinder is displaced downwards by a small distance `y`, the additional volume submerged is `A * y`, where `A` is the cross-sectional area of the cylinder. T...

Answer: To solve the ordinary differential equation (ODE) d²y/dx² + (1/2x) dy/dx + y = 0 using the Frobenius method, we first analyze its singular points. Rewriting the ODE as y'' + (1/2x)y' + y = 0, we identify P(x)=1, Q(x)=1/(2x), R(x)=1. Since x*Q(x)/P(x) = 1/2 and x²*R(x)/P(x) = x², both are analytic at x=0. Thus, x=0 is a regular singular point, making the Frobenius method applicable. We assume a series solution of the form y = Σ(n=0 to ∞) a_n * x^(n+r), where a_0 ≠ 0. The first and second derivat...

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Answer: To determine the temperature distribution u(r, t) within a ball of radius A, subject to specific initial and boundary conditions, we must first define these conditions and then show that the heat conduction equation in spherical coordinates correctly describes the temperature evolution. ### Boundary Conditions The initial and boundary conditions are crucial for solving the partial differential equation. They define the state of the system at the beginning and its interaction with the surround...

Answer: The Fourier series represents a periodic function f(x) with period 2L as f(x) = a₀/2 + Σ[aₙ cos(nπx/L) + bₙ sin(nπx/L)]. For the given function f(x) = π² – x² with a period of 2π, the interval is (-π, π), so L = π in the Fourier series formulas. First, we analyze the symmetry of the function. Since f(-x) = π² – (-x)² = π² – x² = f(x), the function is even. This simplifies the calculation significantly, as all sine coefficients, bₙ, will be zero for an even function over a symmetric interval (-L...

Answer: The deflection u(x, t) of a vibrating string with fixed ends and zero initial velocity is found by solving the one-dimensional wave equation using the method of separation of variables and applying the given initial conditions. The one-dimensional wave equation is given by ∂²u/∂t² = c² ∂²u/∂x², where c is the wave speed. For a string with fixed ends, the boundary conditions are u(0, t) = 0 and u(L, t) = 0. The initial conditions are u(x, 0) = f(x) (given initial deflection) and ∂u/∂t (x, 0) = 0...