Q8
(a) Solve the partial differential equation $$\frac{\partial}{\partial y}\left(\frac{\partial \phi}{\partial x} + \phi\right) + 2x^2y\left(\frac{\partial \phi}{\partial x} + \phi\right) = 0$$ by transforming it to the canonical form. (15 marks) (b) Using Newton's forward difference formula for interpolation, estimate the value of f(2·5) from the following data: x : 1 2 3 4 5 6, f(x) : 0 1 8 27 64 125. (15 marks) (c) Suppose an infinite liquid contains two parallel, equal and opposite rectilinear vortices at a distance 2a. Show that the streamlines relative to the vortex are given by the equation $$\log \frac{x^2 + (y-a)^2}{x^2 + (y+a)^2} + \frac{y}{a} = C,$$ where C is a constant, the origin is the middle point of the join, and the line joining the vortices is the axis of y. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) आंशिक अवकल समीकरण $$\frac{\partial}{\partial y}\left(\frac{\partial \phi}{\partial x} + \phi\right) + 2x^2y\left(\frac{\partial \phi}{\partial x} + \phi\right) = 0$$ को विहित रूप में रूपांतरित करके हल कीजिए। (15 अंक) (b) अंतर्वेशन के लिए न्यूटन के अग्रांतर सूत्र का उपयोग करके निम्नलिखित आँकड़ों से f(2·5) के मान का आकलन कीजिए: x : 1 2 3 4 5 6, f(x) : 0 1 8 27 64 125. (15 अंक) (c) मान लीजिए कि एक अनंत द्रव में दो समानांतर, समान तथा विपरीत सरलरेखीय भ्रमिल 2a की दूरी पर हैं। दर्शाइए कि भ्रमिल के सापेक्ष धारा रेखाएँ समीकरण $$\log \frac{x^2 + (y-a)^2}{x^2 + (y+a)^2} + \frac{y}{a} = C$$ द्वारा दी गई हैं, जहाँ C एक अचर है, मूल-बिंदु जुड़ाव का मध्य बिंदु है, और भ्रमिलों को जोड़ने वाली रेखा y का अक्ष है। (20 अंक)
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How this answer will be evaluated
Approach
Solve all three sub-parts systematically, allocating time proportionally to marks: approximately 18 minutes for part (a) on PDE canonical transformation, 18 minutes for part (b) on Newton's forward interpolation, and 24 minutes for part (c) on vortex dynamics derivation. Begin each part with clear identification of the method, show complete working with proper mathematical notation, and conclude with boxed final answers.
Key points expected
- Part (a): Substitute u = ∂φ/∂x + φ to reduce the PDE to first-order form, then use integrating factor or characteristic method to obtain canonical form
- Part (a): Correctly identify the canonical transformation and solve for φ in terms of arbitrary functions
- Part (b): Construct forward difference table with proper spacing (h=1), identify correct node (x₀=1) and calculate u = (2.5-1)/1 = 1.5
- Part (b): Apply Newton's forward formula with correct binomial coefficients and compute f(2.5) = 0 + 1.5(1) + 0.375(7) + ... leading to value 15.625 or exact verification
- Part (c): Set up complex potential for two opposite vortices at (0,a) and (0,-a) with circulations ±κ
- Part (c): Transform to moving frame with vortex velocity V = κ/(4πa), derive stream function ψ, and obtain streamline equation through algebraic manipulation
- Part (c): Verify the final logarithmic form matches the required expression with constant C
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): Correct substitution u = ∂φ/∂x + φ and proper reduction to canonical form; for (b): Accurate difference table with h=1 and correct u=1.5; for (c): Proper complex potential setup with opposite circulations and correct moving frame transformation | Most setups correct but minor errors in one part, such as wrong h value in (b) or sign error in vortex circulations in (c) | Fundamental setup errors: wrong substitution in (a), incorrect difference formula identification in (b), or incorrect potential for vortex pair in (c) |
| Method choice | 20% | 10 | For (a): Characteristic method or integrating factor for canonical form; for (b): Newton's forward formula with proper binomial coefficient notation; for (c): Complex potential method with Milne-Thomson circle theorem approach or direct stream function derivation | Correct method chosen but inefficient approach or missing standard technique, such as using Lagrange instead of Newton in (b) | Wrong method selected, such as attempting separation of variables in (a), Lagrange interpolation in (b), or velocity superposition without complex potential in (c) |
| Computation accuracy | 20% | 10 | For (a): Correct integration yielding φ = e^{-x}F(y) + e^{-x²y²}G(x) or equivalent; for (b): Precise calculation giving f(2.5)=15.625 (or exact cube verification); for (c): Accurate algebraic manipulation to reach the logarithmic streamline form | Correct approach with arithmetic slips, such as wrong binomial coefficient in (b) leading to approximate answer, or algebraic error in (c) logarithm manipulation | Major computational errors: wrong integration in (a), completely wrong interpolated value in (b), or failure to simplify stream function in (c) |
| Step justification | 20% | 10 | Each mathematical step explicitly justified: why substitution works in (a), why forward differences suit the data in (b), why moving frame eliminates time dependence in (c); proper citation of formulas used | Most steps shown but gaps in reasoning, such as unexplained transition to canonical form or missing justification for frame choice in (c) | Missing crucial steps or 'magic' transitions without explanation, such as unexplained appearance of arbitrary functions or unmotivated formula application |
| Final answer & units | 20% | 10 | All three parts with boxed final answers: general solution for φ in (a), numerical value 15.625 or 125/8 in (b), and derived streamline equation matching the given form exactly in (c); proper dimensional consistency noted | Final answers present but format inconsistent, such as unboxed answers, or minor mismatch in (c) constant definition | Missing final answers, wrong form of solution, or incomplete derivation stopping before required expression in (c) |
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