(a) Using Laplace transform, solve the initial value problem y'' + 2y' + 5y = δ(t-2), y(0) = 0, y'(0) = 0 where δ(t-2) denotes the Dirac delta function. (15 marks)

(b) Using Gauss divergence theorem, evaluate the integral ∬_S (y²î + xz³ĵ + (z-1)²k̂)

Question

(a) Using Laplace transform, solve the initial value problem y'' + 2y' + 5y = δ(t-2), y(0) = 0, y'(0) = 0 where δ(t-2) denotes the Dirac delta function. (15 marks)

(b) Using Gauss divergence theorem, evaluate the integral ∬_S (y²î + xz³ĵ + (z-1)²k̂) · n̂ dS over the region bounded by the cylinder x² + y² = 16 and the planes z = 1 and z = 5. (15 marks)

(c) A particle moves with a central acceleration μ(3/r³ + d²/r⁵) being projected from a distance d at an angle 45° with a velocity equal to that in a circle at the same distance. Prove that the time it takes to reach the centre of force is d²/√(2μ) (2 - π/2). (20 marks)

UPSC Answer Check · Accepted Answer

Solve all three sub-parts systematically, allocating time proportionally to marks: spend ~30% on part (a) Laplace transform problem, ~30% on part (b) Gauss divergence theorem application, and ~40% on part (c) central force motion proof. Begin each part with clear statement of given data, show complete working with proper mathematical notation, and conclude with boxed final answers. For part (c), explicitly state all mechanical assumptions and reference central force theorems used.
- Part (a): Apply Laplace transform to both sides, use L{δ(t-2)} = e^(-2s), solve for Y(s) = e^(-2s)/(s²+2s+5), complete the square to get e^(-2s)/[(s+1)²+4], then apply inverse transform with shift theorem to obtain y(t) = (1/2)e^(-(t-2))sin[2(t-2)]·H(t-2)
- Part (b): Identify divergence of F = ∂(y²)/∂x + ∂(xz³)/∂y + ∂(z-1)²/∂z = 0 + 0 + 2(z-1) = 2(z-1), set up cylindrical coordinates with r: 0-4, θ: 0-2π, z: 1-5, evaluate ∫∫∫ 2(z-1) r dr dθ dz = 2 × 2π × 8 × 8 = 256π
- Part (c): Verify initial conditions satisfy circular orbit velocity v² = μ(3/d² + 1/d²) = 4μ/d², use energy equation and angular momentum conservation, substitute u = 1/r to get differential equation, integrate with proper limits from r=d to r=0, show time integral yields d²/√(2μ) [arccos(0) - arccos(1/√2)] = d²/√(2μ)(2 - π/2)
- Correct handling of Dirac delta function properties and Heaviside step function in part (a)
- Proper application of divergence theorem with closed surface orientation and volume element in cylindrical coordinates for part (b)
- Clear derivation of orbital equation and time integral transformation using substitution u = 1/r for part (c)

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly writes initial conditions for (a), identifies cylinder radius 4 and z-limits for (b), and establishes central force parameters with proper velocity condition for circular orbit in (c); all coordinate systems and variable substitutions are explicitly defined	Most setups are correct but may miss one initial condition in (a), confuse cylinder dimensions in (b), or state velocity condition without derivation in (c)	Major errors in problem setup such as wrong Laplace transform of delta function, incorrect divergence calculation, or fundamental misunderstanding of central force motion setup
Method choice	20%	10	Selects standard Laplace transform method with shift theorem for (a), applies Gauss divergence theorem correctly converting surface to volume integral for (b), and uses Binet's equation or equivalent energy-angular momentum approach with u=1/r substitution for (c)	Correct general methods but may use less efficient approaches or miss optimal substitutions, such as direct integration without shift theorem or Cartesian instead of cylindrical coordinates	Wrong method selection such as undetermined coefficients for (a), direct surface integration ignoring divergence theorem for (b), or attempting Cartesian coordinates for central force in (c)
Computation accuracy	20%	10	Flawless algebraic manipulation including completing the square (s+1)²+4, accurate triple integration yielding 256π, and correct evaluation of inverse trigonometric integrals with precise limits; all arithmetic steps shown	Minor computational errors such as sign errors in completing the square, arithmetic mistakes in integration bounds, or coefficient errors in final time expression that don't affect method validity	Serious computational errors including wrong partial fraction decomposition, incorrect volume element (missing r in cylindrical), or fundamental errors in evaluating the time integral leading to wrong functional form
Step justification	20%	10	Explicitly justifies each key step: shift theorem application with delay justification, divergence calculation showing zero partials, cylindrical coordinate transformation with Jacobian; for (c) proves velocity equals circular orbit speed and justifies the u-substitution with differential relationship	Shows most steps but may omit justification for standard results like L{δ(t-a)} or assume divergence theorem applicability without verifying closed surface; logical flow present but gaps exist	Missing critical justifications such as no mention of Heaviside function emergence, unexplained coordinate changes, or assertion of results without derivation; logical gaps between steps
Final answer & units	20%	10	Presents all three final answers in clean boxed form: y(t) = (1/2)e^(2-t)sin[2(t-2)]H(t-2) or equivalent, 256π for (b), and exactly d²/√(2μ)(2-π/2) for (c); dimensions verified consistent (time, pure number, time respectively)	Correct functional forms but with minor errors in coefficients, or correct numerical answers without proper presentation; units implied but not explicitly checked	Wrong final answers, missing components like Heaviside function, incorrect numerical values, or answers without any dimensional verification; incomplete or illegible final results

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