A heavy string, which is not of uniform density, is hung up from two points. Let T₁, T₂, T₃ be the tensions at the intermediate points A, B, C of the catenary respectively where its inclinations to the horizontal are in arithmetic progression with co

Question

A heavy string, which is not of uniform density, is hung up from two points. Let T₁, T₂, T₃ be the tensions at the intermediate points A, B, C of the catenary respectively where its inclinations to the horizontal are in arithmetic progression with common difference β. Let ω₁ and ω₂ be the weights of the parts AB and BC of the string respectively. Prove that (i) Harmonic mean of T₁, T₂ and T₃ = 3T₂/(1 + 2cos β) (ii) T₁/T₃ = ω₁/ω₂ (20 marks)

Solve the equation: d²y/dx² + (tan x - 3cos x)dy/dx + 2y cos²x = cos⁴x completely by demonstrating all the steps involved. (15 marks)

Evaluate ∫_C F⃗ · dr⃗, where C is an arbitrary closed curve in the xy-plane and F⃗ = (-yî + xĵ)/(x² + y²). (15 marks)

UPSC Answer Check · Accepted Answer

Solve requires complete working with all intermediate steps demonstrated. Structure as: Part I (20 marks) - establish catenary tension relations using ψ = θ - β, θ, θ + β, derive T = T₀ sec ψ, prove harmonic mean identity and tension-weight ratio using vertical equilibrium; Part II (15 marks) - reduce second-order ODE via substitution t = sin x to standard form, find complementary function and particular integral; Part III (15 marks) - identify singularity at origin, apply Green's theorem excluding origin with limiting circle, evaluate residue. Conclude with boxed final answers for each part.
- Catenary: Correct tension formula T = T₀ sec ψ with inclinations ψ₁ = θ - β, ψ₂ = θ, ψ₃ = θ + β in AP
- Catenary: Derivation of T₁ + T₃ = 2T₂ cos β from horizontal equilibrium and harmonic mean manipulation
- Catenary: Vertical equilibrium giving ω₁ = T₂ sin β - T₁ sin β and ω₂ = T₃ sin β - T₂ sin β leading to T₁/T₃ = ω₁/ω₂
- ODE: Substitution t = sin x reducing equation to d²y/dt² - 3dy/dt + 2y = t² with CF = Aeᵗ + Be²ᵗ and PI = ½t² + 3/2t + 7/4
- Line integral: Recognition that ∂Q/∂x - ∂P/∂y = 0 except at origin, Green's theorem with indentation, limit evaluation giving 2π for curves enclosing origin and 0 otherwise
- Complete final answers: (i) H = 3T₂/(1+2cos β), (ii) T₁/T₃ = ω₁/ω₂; y = Aeˢⁱⁿˣ + Be²ˢⁱⁿˣ + ½sin²x + 3/2 sin x + 7/4; integral = 2πn where n = winding number about origin

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies catenary parameterization with inclinations in AP, proper ODE reduction substitution, and singularity analysis for line integral; draws clear diagram for catenary with labeled tensions and angles	Basic setup correct but missing diagram or unclear angle labeling; substitution attempted but not fully justified; recognizes F is singular but analysis incomplete	Wrong angle relationships assumed (e.g., equal angles), incorrect substitution choice for ODE, or fails to identify singularity at origin in line integral
Method choice	20%	10	Uses tension resolution in catenary, harmonic mean algebraic identity, standard ODE reduction to constant coefficients, and Green's theorem with proper indentation technique for singular point	Correct methods chosen but execution lacks elegance; attempts direct integration for ODE or misses indentation argument	Wrong method: attempts direct integration of catenary differential equation, uses variation of parameters unnecessarily, or applies Green's theorem blindly without treating singularity
Computation accuracy	20%	10	Flawless algebra in harmonic mean derivation, correct characteristic roots and undetermined coefficients for ODE, accurate limit evaluation yielding 2π for enclosing curves	Minor algebraic slips in expansion of sec(θ±β) or arithmetic errors in PI coefficients; final answer correct but working contains errors	Major computational errors: incorrect trigonometric identities, wrong roots of auxiliary equation, or evaluates line integral as zero for all curves including those enclosing origin
Step justification	20%	10	Every non-trivial step justified: why inclinations are ψ₁,ψ₂,ψ₃, how vertical equilibrium gives weight relations, justification for substitution choice, rigorous ε-δ argument for indentation limit	Key steps shown but gaps in justification; assumes standard results without citation or skips 'obvious' algebraic manipulations	Unjustified leaps: asserts harmonic mean formula without derivation, claims PI form without verification, or states Green's theorem result without handling discontinuity
Final answer & units	20%	10	All three parts answered completely with exact forms: boxed harmonic mean identity, explicit tension ratio, general solution with arbitrary constants, and piecewise line integral result (2π or 0) with clear condition	Correct answers but not simplified or missing cases (e.g., only gives 2π for line integral without noting origin condition)	Missing answers for parts, wrong final expressions, or numerical values without showing dependence on parameters β, θ

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