Q5
Solve the differential equation: d²y/dx² + 2y = x²e^(3x) + e^x cos 2x (10 marks) Solve the initial value problem: d²y/dx² + 4y = e^(-2x) sin 2x; y(0) = y'(0) = 0 using Laplace transform method. (10 marks) Two rods LM and MN are joined rigidly at the point M such that (LM)² + (MN)² = (LN)² and they are hanged freely in equilibrium from a fixed point L. Let ω be the weight per unit length of both the rods which are uniform. Determine the angle, which the rod LM makes with the vertical direction, in terms of lengths of the rods. (10 marks) If a planet, which revolves around the Sun in a circular orbit, is suddenly stopped in its orbit, then find the time in which it would fall into the Sun. Also, find the ratio of its falling time to the period of revolution of the planet. (10 marks) Show that ∇²[∇·(r⃗/r)] = 2/r⁴, where r⃗ = xî + yĵ + zk̂. (10 marks)
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अवकल समीकरण: d²y/dx² + 2y = x²e^(3x) + e^x cos 2x को हल कीजिए। (10) लाप्लास रूपान्तर विधि का उपयोग करते हुए प्रारम्भिक मान समस्या: d²y/dx² + 4y = e^(-2x) sin 2x; y(0) = y'(0) = 0 को हल कीजिए। (10) दो छड़ें LM व MN बिन्दु M पर दृढ़ता से इस प्रकार जुड़ी हैं कि (LM)² + (MN)² = (LN)² तथा वे स्वतन्त्र रूप से साम्यावस्था में स्थिर बिन्दु L पर टंगी हैं। माना कि दोनों एकसमान छड़ों का प्रति एकांक लम्बाई, भार ω है। छड़ LM का उद्वधर दिशा के साथ बने कोण को छड़ों की लम्बाई के रूप में ज्ञात कीजिए। (10) यदि एक ग्रह, जो सूर्य के परितः वृत्तीय कक्षा में परिभ्रमण करता है, अचानक अपनी कक्षा में रोक दिया जाता है, तो वह समय, जिसमे वह सूर्य में गिर जाएगा, ज्ञात कीजिए। इसके गिरने के समय का ग्रह के परिभ्रमण आवर्तकाल से अनुपात भी ज्ञात कीजिए। (10) दर्शाइए कि ∇²[∇·(r⃗/r)] = 2/r⁴, जहाँ r⃗ = xî + yĵ + zk̂ है। (10)
Directive word: Solve
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How this answer will be evaluated
Approach
Solve demands complete analytical solutions with rigorous mathematical derivation. Structure as: (1) Differential equation using undetermined coefficients/operator method, (2) Laplace transform with partial fractions and inversion, (3) Static equilibrium with virtual work or moment balance for LMN triangle configuration, (4) Central force motion converting circular to radial free-fall using Kepler's laws, (5) Vector calculus with proper coordinate treatment of r/|r|. Each part requires distinct mathematical machinery.
Key points expected
- Part (a): Correct complementary function y_c = A cos(√2 x) + B sin(√2 x) and particular integral using operator method or undetermined coefficients for both x²e^(3x) and e^x cos 2x terms
- Part (b): Proper Laplace transform application with ℒ{e^(-2x) sin 2x} = 2/((s+2)²+4), correct partial fraction decomposition, and inverse transform satisfying y(0) = y'(0) = 0
- Part (c): Recognition that angle at M is 90°, use of virtual work principle or moment equilibrium about L, expressing tan θ = (MN)²/(LM·LN) or equivalent in terms of given lengths
- Part (d): Conversion of circular orbit to degenerate ellipse with semi-major axis a = R/2, application of Kepler's third law giving fall time T/4√2, ratio 1/(4√2) or √2/8
- Part (e): Correct computation of ∇·(r⃗/r) = 2/r, then ∇²(2/r) = 0 for r ≠ 0, with proper handling of singularity or verification via direct Cartesian differentiation
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies equation type (linear ODE with constant coefficients), writes proper auxiliary equation m²+2=0, sets up Laplace transform with correct initial conditions, recognizes right-angled triangle geometry for rods, applies vis-viva or energy equation for orbital decay, and uses correct vector identities for divergence and Laplacian | Most setups correct but minor errors in auxiliary equation roots, initial condition application, or geometric configuration; may confuse r⃗/r with unit vector properties | Fundamental setup errors: wrong auxiliary equation, incorrect Laplace transform definition, treats LMN as general triangle not right-angled, uses free-fall acceleration g instead of gravitational dynamics, or confuses ∇·r⃗ with ∇·(r⃗/r) |
| Method choice | 20% | 10 | Uses operator method (1/(D²+2)) for particular integral with proper exponential shift theorem, standard Laplace inversion techniques, virtual work or Lagrangian for statics, Kepler's laws for orbital mechanics, and direct Cartesian or spherical coordinate verification for vector identity | Uses variation of parameters instead of operator method (inefficient but valid), attempts convolution theorem unnecessarily for Laplace, uses force balance rather than energy methods, or switches to spherical coordinates without justification | Inappropriate methods: undetermined coefficients without operator shift for exponential terms, attempts numerical methods for Laplace, uses incorrect static equilibrium equations, treats as simple harmonic motion for orbital fall, or attempts index notation without proper tensor calculus |
| Computation accuracy | 20% | 10 | Flawless algebraic manipulation: correct partial fraction coefficients, accurate trigonometric identities in particular integral, precise angle determination from length ratios, exact expression T√2/8 for fall time ratio, and verified ∇²(1/r) = 0 leading to 2/r⁴ | Minor computational slips: sign errors in particular integral coefficients, arithmetic errors in partial fractions, incorrect algebraic simplification of angle formula, or off-by-factor errors in orbital period calculation | Major computational failures: incorrect integration by parts for particular integral, systematic errors in Laplace inversion, wrong trigonometric resolution for rod angles, confused energy conservation in orbital problem, or elementary errors in partial derivatives |
| Step justification | 20% | 10 | Explicitly states shift theorem for operator D→D+3, shows partial fraction decomposition steps, justifies virtual displacement or moment arm choices, derives vis-viva from energy conservation, and proves vector identity through component-wise verification with intermediate steps displayed | Some steps shown but skips key justifications: assumes form of particular integral without explanation, omits partial fraction working, asserts equilibrium conditions without free-body diagrams, or jumps to final vector result | Minimal working shown: writes answer without derivation, states results without intermediate steps, or presents disconnected formulae without logical flow between steps |
| Final answer & units | 20% | 10 | Complete general solution with arbitrary constants for ODE, explicit y(t) satisfying all initial conditions, angle expressed as arctan function of LM, MN, LN, dimensionless ratio √2/8 or equivalent simplified form, and verified identity with proper domain specification (r ≠ 0) | Correct forms but incomplete: misses arbitrary constants, doesn't verify initial conditions are satisfied, leaves angle in implicit form, or doesn't simplify ratio to standard form | Missing or wrong answers: no particular integral, incorrect final y(t), unsolved for angle, wrong time expression, or incorrect vector identity result without verification |
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