Q7
(a) State uniqueness theorem for the existence of unique solution of the initial value problem dy/dx = f(x, y), y(x₀) = y₀ in the rectangular region R: |x - x₀| ≤ a, |y - y₀| ≤ b. Test the existence and uniqueness of the solution of the initial value problem dy/dx = 2√y, y(1) = 0, in a suitable rectangle R. If more than one solution exist, then find all the solutions. (15 marks) (b) A heavy particle hanging vertically from a fixed point by a light inextensible string of length l starts to move with initial velocity u in a circle so as to make a complete revolution in a vertical plane. Show that the sum of tensions at the ends of any diameter is constant. (15 marks) (c) State Stokes' theorem and verify it for the vector field F⃗ = xyî + yzĵ + zxk̂ over the surface S, which is the upwardly oriented part of the cylinder z = 1 - x², for 0 ≤ x ≤ 1, -2 ≤ y ≤ 2. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) आयतीय क्षेत्र R: |x - x₀| ≤ a, |y - y₀| ≤ b में प्रारंभिक मान समस्या dy/dx = f(x, y), y(x₀) = y₀ के अद्वितीय हल के अस्तित्व के लिए अद्वितीयता प्रमेय का कथन लिखिए। एक उपयुक्त आयत R में प्रारंभिक मान समस्या dy/dx = 2√y, y(1) = 0 के हल के अस्तित्व और अद्वितीयता का परीक्षण कीजिए। यदि एक से अधिक हल मौजूद हैं, तो सभी हलों को ज्ञात कीजिए। (15 अंक) (b) लंबाई l की एक हल्की अवितान्य डोरी द्वारा एक नियत बिंदु से उर्ध्वाधर लटका हुआ एक भारी कण प्रारंभिक वेग u के साथ एक वृत्त में घूमना शुरू करता है ताकि एक उर्ध्वाधर समतल में एक पूर्ण परिक्रमण कर सके। दर्शाइए कि किसी भी व्यास के सिरों पर तनावों का योग अचर है। (15 अंक) (c) स्टोक्स प्रमेय का कथन लिखिए तथा इसको सदिश क्षेत्र F⃗ = xyî + yzĵ + zxk̂ के लिए, पृष्ठ S पर जो कि बेलन z = 1 - x²; 0 ≤ x ≤ 1, -2 ≤ y ≤ 2 का उपरिमुखी अभिविन्यस्त भाग है, सत्यापित कीजिए। (20 अंक)
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How this answer will be evaluated
Approach
Solve all three sub-parts systematically, allocating approximately 25-30% time to part (a) on existence-uniqueness theorem, 25-30% to part (b) on vertical circular motion dynamics, and 40-45% to part (c) on Stokes' theorem verification as it carries the highest marks. Begin each part with precise statement of relevant theorems, followed by rigorous mathematical working, and conclude with clear final answers.
Key points expected
- Part (a): Correct statement of Picard's existence and uniqueness theorem with conditions on f(x,y) and ∂f/∂y being continuous in R
- Part (a): Identification that f(x,y) = 2√y is continuous but ∂f/∂y = 1/√y is unbounded at y=0, violating uniqueness condition
- Part (a): Derivation of two distinct solutions: y=0 and y=(x-1)², proving non-uniqueness
- Part (b): Application of energy conservation and tension formula T = mv²/l + mgcosθ for vertical circular motion
- Part (b): Proof that T(θ) + T(θ+π) = 2mv₀²/l = constant, using velocity relation from energy equation
- Part (c): Correct statement of Stokes' theorem: ∮_C F⃗·dr⃗ = ∬_S (∇×F⃗)·n̂ dS
- Part (c): Computation of curl ∇×F⃗ = -yî - zĵ - xk̂ and proper parametrization of cylindrical surface
- Part (c): Evaluation of both line integral around boundary curves and surface integral, showing equality
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Precisely states Picard's theorem with both continuity conditions for (a); correctly identifies forces and draws free-body diagram for (b); accurately states Stokes' theorem with proper orientation for (c). Defines all variables and regions clearly. | States theorems with minor omissions (e.g., misses ∂f/∂y condition in (a) or orientation in (c)); basic setup correct but lacks precision in defining domains or variables. | Misstates theorems or confuses existence with uniqueness; incorrect force identification in (b); wrong orientation or surface description in (c); missing crucial conditions. |
| Method choice | 20% | 10 | Selects optimal approach: Lipschitz condition analysis for (a); energy method with tension formula for (b); direct computation of both sides for (c) with efficient parametrization. Recognizes need to check boundary behavior in (a). | Uses correct general methods but with suboptimal choices (e.g., attempts Picard iteration unnecessarily in (a), or uses Cartesian instead of cylindrical coordinates in (c)). | Wrong method entirely (e.g., tries integrating factor for (a), uses projectile motion for (b), or applies Gauss divergence theorem for (c)); confused or missing methodology. |
| Computation accuracy | 20% | 10 | Flawless calculations: correct partial derivatives and limit analysis in (a); accurate algebraic manipulation showing tension sum independence from θ in (b); precise curl computation, surface normal, and integral evaluation in (c) with correct final numerical equality. | Minor computational errors (sign errors in curl components, arithmetic slips in tension algebra, or incorrect limits in surface integral) that don't completely invalidate the solution. | Major computational errors: wrong derivatives, incorrect curl calculation, fundamental errors in line/surface integral setup, or algebraic mistakes leading to wrong conclusions. |
| Step justification | 20% | 10 | Every step rigorously justified: explains why continuity fails for uniqueness in (a); clearly derives velocity from energy conservation in (b); shows parametrization validity and orientation consistency in (c). Logical flow with explicit theorem citations. | Most steps shown but with gaps in justification (e.g., assumes existence of solutions without verification, skips energy derivation, or asserts orientation without explanation). | Missing crucial justifications (no explanation for multiple solutions in (a), asserts tension formula without derivation in (b), or verifies Stokes' theorem without showing work for both sides). |
| Final answer & units | 20% | 10 | Complete answers: explicitly states both solutions y=0 and y=(x-1)² for (a); clearly proves constant sum 2mv₀²/l (or equivalent) for (b); shows exact numerical equality of both integrals in (c). Proper dimensional analysis where applicable. | Correct final forms but incomplete (finds only one solution in (a), states result without proof in (b), or shows partial verification in (c)). | Missing or wrong final answers; fails to identify multiple solutions in (a), incorrect constant value in (b), or unequal integrals in (c); no conclusion drawn. |
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