Paper II — Q8
(a) Find the characteristics of the partial differential equation p² + q² = 2; p ≡ ∂z/∂x, q ≡ ∂z/∂y and determine the integral…
Find the characteristics of the partial differential equation p² + q² = 2; p ≡ ∂z/∂x, q ≡ ∂z/∂y and determine the integral surface which passes through x = 0, z = y. 15 marks
Find the constant p and error term for the quadrature formula ∫ₓ₀^x₁ f(x)dx = (h/2)(f₀ + f₁) + ph²(f'₀ - f'₁) where x₀ + h = x₁, f₀ = f(x₀), f₁ = f(x₁) and prime (') represents derivative with respect to x. Hence deduce the composite rule for integrating ∫ₐ^b f(x)dx, a = x₀ < x₁ < ... < x_N = b 15 marks
A particle of mass m moves in a force field of potential V(r) = -k cosθ/r², k is constant Find the Hamiltonian and the Hamilton's equations in spherical polar coordinates (r, θ, φ). 10 marks
Consider the Lagrangian L = mẋẏ - mω₀²xy where m and ω₀ are constants. Find the Hamiltonian and Hamilton's equations of motion. Identify the system. 10 marks
हिंदी में प्रश्न पढ़ें
आंशिक अवकल समीकरण p² + q² = 2; p ≡ ∂z/∂x, q ≡ ∂z/∂y के अभिलक्षण (कैरेक्टरिस्टिक्स) ज्ञात कीजिए और x = 0, z = y से होकर जाने वाला समाकल पृष्ठ (सरफेस) प्राप्त कीजिए। (15 अंक)
क्षेत्रकलन-सूत्र ∫ₓ₀^x₁ f(x)dx = (h/2)(f₀ + f₁) + ph²(f'₀ - f'₁) जहाँ x₀ + h = x₁, f₀ = f(x₀), f₁ = f(x₁) है और प्राइम ('), x के सापेक्ष अवकलज को निर्दिष्ट करता है, के लिए अचर p और त्रुटि-पद ज्ञात कीजिए। अतः समाकलन ∫ₐ^b f(x)dx, a = x₀ < x₁ < ... < x_N = b का मान ज्ञात करने के लिए संयुक्त नियम का निगमन कीजिए। (15 अंक)
विभव V(r) = -k cosθ/r², k अचर है के एक बल क्षेत्र में द्रव्यमान m का एक कण गतिमान है। गोलीय ध्रुवीय निर्देशांकों (r, θ, φ) में हैमिल्टोनियन और हैमिल्टन का समीकरण ज्ञात कीजिए। (10 अंक)
लैग्रांजी L = mẋẏ - mω₀²xy, जहाँ m और ω₀ अचर हैं, का विचार कीजिए। हैमिल्टोनियन और गति का हैमिल्टन समीकरण ज्ञात कीजिए। तंत्र (सिस्टम) की पहचान बताइए। (10 अंक)
Model answer
Drafted with AI by UPSC Answer Check against this question's marking rubric, to the expected length. UPSC does not publish answers for Mains — this is one way to score well, not an official key. How we build rubrics and answers.
(a)
For F(x,y,z,p,q)=p²+q²−2=0, the characteristic equations are dx/F_p = dy/F_q = dz/(pF_p+qF_q) = −dp/(F_x+pF_z) = −dq/(F_y+qF_z). Here F_p=2p, F_q=2q, F_x=F_y=F_z=0. Hence dx/(2p)=dy/(2q)=dz/(2(p²+q²)), and dp=0, dq=0. So p=a, q=b with a²+b²=2. Taking s as parameter, dx/ds=2a, dy/ds=2b, dz/ds=2(a²+b²)=4. The initial curve is x=0, z=y. Parametrize it by y=t, z=t. The strip condition dz=p dx+q dy gives, on x=0, dz=q dy. Since dz=dy, q=1. Then a²+1=2 ⇒ a=±1. Thus two branches arise.
For p=1, q=1: dx/ds=2, dy/ds=2, dz/ds=4. From (0,t,t), x=2s, y=t+2s, z=t+4s, so z=x+y. For p=−1, q=1: x=−2s, y=t+2s, z=t+4s, so z=y−x. Therefore the Cauchy problem has two integral surfaces: z=x+y and z=y−x; if the branch p>0 is selected, then z=x+y.
(b)
Put x=x₀+th. Then ∫_x₀^x₁ f(x)dx = h∫₀¹ f(x₀+ht)dt. Using Taylor expansion, I = h f₀ + h²/2 f₀′ + h³/6 f₀″ + h⁴/24 f₀‴ + h⁵/120 f₀⁗ + … The given formula is Q = h/2(f₀+f₁)+p h²(f₀′−f₁′). Expanding f₁ and f₁′ about x₀, Q = h f₀ + h²/2 f₀′ + (1/4−p)h³ f₀″ + (1/12−p/2)h⁴ f₀‴ + (1/48−p/6)h⁵ f₀⁗ + … Matching the h³ coefficient with I gives 1/4−p=1/6 ⇒ p=1/12. Then the h⁴ terms also match. The h⁵ coefficient in Q is 1/144, while in I it is 1/120. Therefore I−Q = (1/120−1/144)h⁵ f₀⁗ + … = h⁵/720 f₀⁗ + … By the extended mean-value theorem, if f∈C⁴, the error term is h⁵/720 f⁽⁴⁾(ξ), x₀<ξ<x₁. Thus ∫_x₀^x₁ f dx = (h/2)(f₀+f₁)+(h²/12)(f₀′−f₁′) + h⁵/720 f⁽⁴⁾(ξ). For the composite rule, with h=(b−a)/N and nodes x_i, ∫_a^b f dx = h/2[f₀+f_N+2Σ_i=1^N−1f_i] + h²/12(f₀′−f_N′) + (b−a)h⁴/720 f⁽⁴⁾(ξ), where ξ∈(a,b).
(c)(i)
In spherical polar coordinates, T = (m/2)(ṙ²+r²θ̇²+r²sin²θ φ̇²), V = −k cosθ/r². The momenta are p_r = mṙ, p_θ = m r²θ̇, p_φ = m r²sin²θ φ̇. Hence the Hamiltonian is H = p_r²/(2m)+p_θ²/(2m r²)+p_φ²/(2m r²sin²θ) − k cosθ/r². Hamilton’s equations are ṙ = ∂H/∂p_r = p_r/m, θ̇ = ∂H/∂p_θ = p_θ/(m r²), φ̇ = ∂H/∂p_φ = p_φ/(m r²sin²θ), ṗ_r = −∂H/∂r = p_θ²/(m r³)+p_φ²/(m r³sin²θ) − 2k cosθ/r³, ṗ_θ = −∂H/∂θ = p_φ² cosθ/(m r²sin³θ) − k sinθ/r², ṗ_φ = −∂H/∂φ = 0. Thus p_φ is constant. Validity requires r>0, 0<θ<π, and sinθ≠0 where p_φ is expressed.
(c)(ii)
For L=mẋẏ−mω₀²xy, p_x = ∂L/∂ẋ = mẏ, p_y = ∂L/∂ẏ = mẋ. Thus ẋ=p_y/m, ẏ=p_x/m. The Hamiltonian is H = p_xẋ+p_yẏ−L = p_x p_y/m + mω₀²xy. Hamilton’s equations are ẋ = ∂H/∂p_x = p_y/m, ẏ = ∂H/∂p_y = p_x/m, ṗ_x = −∂H/∂x = −mω₀²y, ṗ_y = −∂H/∂y = −mω₀²x. Differentiating again, ẍ = −ω₀²x, ÿ = −ω₀²y. So the system describes two uncoupled simple harmonic oscillators of angular frequency ω₀ along x and y, written in a nonstandard cross-momentum Lagrangian form.
How this answer will be evaluated
Framework: Charpit's Method for Non-linear PDEs
What earns full marks: Rigorous derivations with all steps shown and correct final results for all parts.
Evaluation rubric
Each sub-part is marked on its own, against the marks and word limit printed on the paper.
- (a) Find characteristics and the specific integral surface for the PDE p² + q² = 2. 15 marks
derive— given → assumptions → stepwise derivation → result → check
Must cover
- Characteristic equations for F = p² + q² − 2
- dp = 0, dq = 0 ⇒ p = a, q = b with a² + b² = 2; dz/ds = 4
- Initial curve x = 0, z = y with strip condition ⇒ q = 1, a = ±1
- Integral surfaces z = x + y (p = 1) and z = y − x (p = −1)
Loses marks
- Incorrect setup of Charpit's equations
- Failure to apply the boundary condition
Earns more
- Correct identification of p and q
- Clear step-by-step integration of auxiliary equations
Extra mark
- Verification of the final surface in the PDE
- (b) Determine constant p and error term for the quadrature formula, then deduce the composite rule. 15 marks
derive— given → assumptions → stepwise derivation → result → check
Must cover
- Taylor expansion of I and Q in powers of h
- Match h³: 1/4 − p = 1/6 ⇒ p = 1/12
- h⁵ mismatch: I − Q = h⁵/720 f⁗(ξ) (error term)
- Composite rule: (h/2)[f₀ + f_N + 2Σfᵢ] + (h²/12)(f₀′ − f_N′) + ((b−a)h⁴/720)f⁗(ξ)
Loses marks
- Incorrect Taylor expansion terms
- Failure to sum over the composite intervals
Earns more
- Correct identification of the modified trapezoidal rule
- Clear derivation of the composite summation
Extra mark
- Comparison with standard trapezoidal rule error
- (c)(i) Find the Hamiltonian and Hamilton's equations for a particle in the given potential field. 10 marks
derive— given → assumptions → stepwise derivation → result → check
Must cover
- T in spherical polars; momenta p_r, p_θ, p_φ
- H = p_r²/2m + p_θ²/(2mr²) + p_φ²/(2mr²sin²θ) − k cosθ/r²
- Hamilton equations for coordinates and momenta
- ṗ_φ = 0 (cyclic); validity r > 0, 0 < θ < π
Loses marks
- Incorrect kinetic energy expression in spherical coordinates
- Missing terms in the Hamiltonian
Earns more
- Correct expression for conjugate momenta
- Clear separation of radial and angular components
Extra mark
- Identification of conserved quantities (e.g., angular momentum)
- (c)(ii) Find the Hamiltonian and equations of motion for the given Lagrangian and identify the system. 10 marks
derive— given → assumptions → stepwise derivation → result → check
Must cover
- p_x = mẏ, p_y = mẋ; ẋ = p_y/m, ẏ = p_x/m
- H = p_x p_y/m + mω₀²xy (Legendre transformation)
- Hamilton's equations; ẍ = −ω₀²x, ÿ = −ω₀²y
- Identify: two uncoupled simple harmonic oscillators of frequency ω₀
Loses marks
- Incorrect Legendre transformation
- Failure to identify the physical system
Earns more
- Correct identification of the system as two uncoupled simple harmonic oscillators
- Clear derivation of the equations of motion
Extra mark
- Analysis of the system's stability or normal modes
What "Solve" is asking you to do
Choose the method, then carry it through to a final answer. Identifying what kind of problem this is and why that method applies is the first thing marked; a correct figure arrived at invisibly earns almost nothing.
Structure that answers it
Given data and what is required → method chosen, with the reason it applies → set-up (equation, circuit, free body, trial balance) → working, step by step → answer with units and any condition of validity
Where marks are lost
Doing the middle steps mentally and writing only the result. In mathematics papers, a further loss comes from giving a decimal where the exact value in surds or fractions was wanted, or from skipping the justification a part explicitly asks for.
Practice this exact question
Write your answer and it is marked point by point against the model answer above — what you covered, what you missed, what you got wrong.
Evaluate my answer →More from Mathematics 2025 Paper II
- Q5 (a) Find the solution of the equation (D² + DD' - 2D'²)z = ysin x, where D ≡ (∂)/(∂ x) an…
- Q6 (a) Solve (∂² u)/(∂ x²) + (∂² u)/(∂ y²) = 0 for a rectangular plate subject to the bounda…
- Q7 (a) Find the complete integral of z(p²-q²) = x-y; p≡∂z/∂x, q≡∂z/∂y. (15 marks) (b) Find t…
- Q8 (a) Find the characteristics of the partial differential equation p² + q² = 2; p ≡ ∂z/∂x,…