Q6
(a) Find the surface passing through the two lines z = x = 0 and z-1 = x-y = 0, and satisfying the partial differential equation ∂²z/∂x² - 4∂²z/∂x∂y + 4∂²z/∂y² = 0. (15 marks) (b) Solve the system of linear equations 7x₁ - x₂ + 2x₃ = 11, 2x₁ + 8x₂ - x₃ = 9, x₁ - 2x₂ + 9x₃ = 7 correct up to 4 significant figures by the Gauss-Seidel iterative method. Take initially guessed solution as x₁ = x₂ = x₃ = 0. (15 marks) (c) A mechanical system with 2 degrees of freedom has the Lagrangian L = ½m(ẋ² + ẏ²) - ½m(w₁²x² + w₂²y²) + kxy where m, w₁, w₂, k are constants. Find the parameter θ so that under the transformation x = q₁ cos θ - q₂ sin θ, y = q₁ sin θ + q₂ cos θ the Lagrangian in terms of q₁, q₂ will not contain the product term q₁q₂. Find the Lagrange's equations w.r.t. q₁ and q₂ independent of parameter θ. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) दो रेखाओं z = x = 0 तथा z-1 = x-y = 0 से होकर जाने वाला और आंशिक अवकल समीकरण ∂²z/∂x² - 4∂²z/∂x∂y + 4∂²z/∂y² = 0 को संतुष्ट करने वाला पृष्ठ ज्ञात कीजिए। (15 अंक) (b) गॉस-सीडेल पुनरावृत्ति विधि से रैखिक समीकरण निकाय 7x₁ - x₂ + 2x₃ = 11, 2x₁ + 8x₂ - x₃ = 9, x₁ - 2x₂ + 9x₃ = 7 का 4 सार्थक अंकों तक सही हल ज्ञात कीजिए। आरंभिक अनुमानित हल x₁ = x₂ = x₃ = 0 लीजिए। (15 अंक) (c) स्वतंत्रता की कोटि 2 के एक यांत्रिक तंत्र का लैग्रांजियन L = ½m(ẋ² + ẏ²) - ½m(w₁²x² + w₂²y²) + kxy है, जहाँ m, w₁, w₂, k अचर हैं। वह प्राचल θ ज्ञात कीजिए, जिसके लिए रूपांतरण x = q₁ cos θ - q₂ sin θ, y = q₁ sin θ + q₂ cos θ के अंतर्गत q₁, q₂ के पदों में लैग्रांजियन में गुणन पद q₁q₂ नहीं होगा। प्राचल θ से स्वतंत्र, q₁ तथा q₂ के सापेक्ष लग्रांज समीकरणों को ज्ञात कीजिए। (20 अंक)
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How this answer will be evaluated
Approach
Solve all three sub-parts systematically: for (a) use Monge's auxiliary equations to find the general solution and apply boundary conditions; for (b) apply Gauss-Seidel iteration with proper convergence check to 4 significant figures; for (c) transform coordinates, eliminate cross terms by choosing θ = ½ tan⁻¹(2k/(m(w₁²-w₂²))), then derive uncoupled Lagrange equations. Allocate time proportionally: ~30% to (a), ~30% to (b), and ~40% to (c) given mark distribution.
Key points expected
- For (a): Recognize PDE as parabolic type (B²-AC=0), use characteristic coordinates ξ=y+2x, η=y to reduce to canonical form and obtain general solution z = f(y+2x) + x·g(y+2x)
- For (a): Apply boundary conditions on lines z=x=0 and z-1=x-y=0 to determine arbitrary functions f and g, yielding specific surface equation
- For (b): Verify diagonal dominance of coefficient matrix, rewrite equations in iterative form, execute Gauss-Seidel iterations until convergence to 4 significant figures
- For (c): Transform kinetic and potential energy terms using rotation matrix, collect coefficients of q₁q₂ and set to zero to find tan(2θ) = 2k/(m(w₁²-w₂²))
- For (c): Obtain diagonalized Lagrangian L = ½m(Ω₁²q̇₁² + Ω₂²q̇₂²) - ½m(Ω₁²q₁² + Ω₂²q₂²) and derive two independent harmonic oscillator equations
- For (c): Express final frequencies Ω₁² = ω₁²cos²θ + ω₂²sin²θ - (k/m)sin2θ and Ω₂² = ω₁²sin²θ + ω₂²cos²θ + (k/m)sin2θ in terms of original parameters
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly classifies PDE in (a) as parabolic with discriminant zero; verifies diagonal dominance for Gauss-Seidel convergence in (b); properly constructs rotation transformation matrix in (c) with correct Jacobian for coordinate change | Identifies PDE type correctly but misapplies characteristic method; attempts Gauss-Seidel without convergence check; sets up transformation but makes algebraic errors in Jacobian or metric | Wrong PDE classification (elliptic/hyperbolic); uses Jacobi instead of Gauss-Seidel or fails to rearrange equations; incorrect transformation setup or ignores velocity transformation |
| Method choice | 20% | 10 | Uses Monge's method for (a) with proper auxiliary equations; applies optimized Gauss-Seidel with error estimation for (b); employs normal mode diagonalization technique for (c) with systematic elimination of cross terms | Uses separation of variables or D'Alembert's approach for (a); applies iteration without stopping criterion; attempts diagonalization but with ad hoc parameter choice | Attempts separation of variables for non-separable PDE; uses direct matrix inversion for (b); brute force expansion without recognizing diagonalization structure |
| Computation accuracy | 20% | 10 | Exact integration constants in (a); iterates to 4 significant figure precision (typically 5-6 iterations) with x₁≈1.433, x₂≈0.896, x₃≈0.815 in (b); correct algebraic simplification leading to clean frequency expressions in (c) | Minor arithmetic errors in boundary condition application; 3-4 correct significant figures or correct iterations with rounding errors; correct θ formula but messy final algebra | Major integration errors; divergent iterations or wrong final values; incorrect trigonometric identities leading to wrong θ or uncoupled equations still containing cross terms |
| Step justification | 20% | 10 | Explicitly justifies characteristic choice via dx/1 = dy/(-2) = dz/0; shows each iteration's error estimate; proves why cross term vanishes and verifies resulting equations are indeed θ-independent | States Monge's equations without derivation; lists iterations without convergence analysis; states result without verifying independence from θ | No justification for method steps; missing iterations or unexplained jumps; no verification that final equations are θ-independent |
| Final answer & units | 20% | 10 | Clean closed-form surface equation for (a); properly rounded 4-significant-figure answers with iteration count for (b); explicit θ formula and manifestly θ-independent Lagrange equations with physical interpretation of normal modes for (c) | Correct form but unsimplified; correct values with minor rounding issues; correct structure but θ-dependence remains implicit | Incomplete or wrong final expressions; unconverged or wrong numerical values; missing final equations or wrong structure |
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