(a) By eliminating the arbitrary functions f and g from z = f(x² - y) + g(x² + y), form partial differential equation. (10 marks)

(b) Given dy/dx = (y² - x)/(y² + x) with initial condition y = 1 at x = 0. Find the value of y for x = 0.4 by Euler's m

Question

(a) By eliminating the arbitrary functions f and g from z = f(x² - y) + g(x² + y), form partial differential equation. (10 marks)

(b) Given dy/dx = (y² - x)/(y² + x) with initial condition y = 1 at x = 0. Find the value of y for x = 0.4 by Euler's method, correct to 4 decimal places, taking step length h = 0.1. (10 marks)

(c) Evaluate, using the binary arithmetic, the following numbers in their given system:
(i) (634.235)₈ - (132.223)₈
(ii) (7AB.432)₁₆ - (5CA.D61)₁₆ (10 marks)

(d) A planet of mass m is revolving around the sun of mass M. The kinetic energy T and the potential energy V of the planet are given by T = ½m(ṙ² + r²θ̇²) and V = GMm(1/2a - 1/r), where (r, θ) are the polar coordinates of the planet at time t, G is the gravitational constant and 2a is the major axis of the ellipse (the path of the planet). Find the Hamiltonian and the Hamilton equations of the planet's motion. (10 marks)

(e) In a fluid motion, there is a source of strength 2m placed at z = 2 and two sinks of strength m are placed at z = 2 + i and z = 2 - i. Find the streamlines. (10 marks)

UPSC Answer Check · Accepted Answer

Solve each sub-part systematically with clear working: for (a) apply partial differentiation to eliminate arbitrary functions; for (b) execute Euler's method with 4 iterations; for (c)(i)-(ii) perform octal and hexadecimal subtraction with borrowing; for (d) construct Hamiltonian from T-V and derive canonical equations; for (e) use complex potential for source-sink superposition. Allocate approximately 15% time to (a), 20% to (b), 20% to (c), 25% to (d), and 20% to (e), ensuring all numerical answers are boxed with correct precision.
- For (a): Correctly identify arguments u = x² - y and v = x² + y, compute partial derivatives p = ∂z/∂x and q = ∂z/∂y, then eliminate f' and g' to obtain PDE: x(∂z/∂x) = 2(∂z/∂y) or equivalent second-order form
- For (b): Apply Euler's formula y_{n+1} = y_n + hf(x_n, y_n) for 4 steps (x=0 to 0.4), showing each iteration with h=0.1, final answer y(0.4) ≈ 1.4615 (to 4 decimal places)
- For (c)(i): Perform octal subtraction (634.235)₈ - (132.223)₈ = (502.012)₈ with proper borrowing across radix point
- For (c)(ii): Perform hexadecimal subtraction (7AB.432)₁₆ - (5CA.D61)₁₆ = (1E0.871)₁₆ handling borrow from 16's complement
- For (d): Form Hamiltonian H = T + V = ½m(p_r²/m² + p_θ²/(m²r²)) + GMm(1/2a - 1/r), derive ṙ = ∂H/∂p_r, θ̇ = ∂H/∂p_θ, ṗ_r = -∂H/∂r, ṗ_θ = -∂H/∂θ = 0
- For (e): Construct complex potential W = 2m ln(z-2) - m ln(z-2-i) - m ln(z-2+i), extract stream function ψ = Im(W), obtain streamline equation or implicit form

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies all initial conditions, variable substitutions, and problem structures: proper arguments u,v for (a), Euler setup with h=0.1 for (b), radix alignment for (c), canonical momenta for (d), and source-sink locations in complex plane for (e)	Most setups correct but minor errors in one sub-part: wrong step size, misaligned radix points, or incorrect canonical momentum definitions	Fundamental setup errors in multiple parts: incorrect elimination approach, wrong Euler formula, confused number bases, or invalid Hamiltonian construction
Method choice	20%	10	Selects optimal methods: Charpit's approach for PDE elimination, forward Euler for ODE, 8's/16's complement or direct borrowing for arithmetic, Legendre transform for Hamiltonian, and complex potential superposition for fluid flow	Methods mostly appropriate but suboptimal: missing efficiency in arithmetic or slightly roundabout PDE elimination	Inappropriate methods chosen: using Runge-Kutta instead of Euler, decimal instead of binary arithmetic, or Lagrangian instead of Hamiltonian formalism
Computation accuracy	20%	10	All calculations precise: correct second-order PDE obtained, Euler iterations accurate to 4 decimal places (y₁=1.1000, y₂=1.1918, y₃=1.2780, y₄≈1.4615), exact octal/hexadecimal results, correct Hamilton equations, proper streamline derivation	Minor computational slips: rounding errors in Euler beyond 4th decimal, single borrowing error in arithmetic, or sign error in one Hamilton equation	Significant computational errors: wrong PDE order, completely wrong Euler trajectory, multiple base conversion errors, or incorrect Hamiltonian sign
Step justification	20%	10	Every non-trivial step explicitly justified: shows cross-multiplication for PDE elimination, displays all 4 Euler iterations with f(x,y) evaluations, writes borrow chains for subtraction, derives p_r = mṙ and p_θ = mr²θ̇, explains stream function from complex potential	Most steps shown but some gaps: missing one Euler iteration, skipping borrow details, or assuming canonical momenta without derivation	Major logical gaps: jumps from z=f+g to final PDE without working, presents only final Euler answer, or states Hamiltonian without derivation
Final answer & units	20%	10	All answers boxed with correct precision: PDE in standard form, y(0.4)=1.4615, (502.012)₈ and (1E0.871)₁₆, four Hamilton equations clearly stated, streamline equation ψ = constant or explicit y(x) form	Answers present but format issues: missing subscripts for bases, insufficient decimal places, or implicit rather than explicit streamline form	Missing or wrong final answers: no PDE stated, only intermediate Euler values, arithmetic without final result, incomplete Hamilton equations, or no streamline equation

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