Q7
(a) Find the general solution of the partial differential equation $$(D^2 + DD' - 6D'^2)z = x^2 \sin(x+y)$$ where $D \equiv \frac{\partial}{\partial x}$ and $D' \equiv \frac{\partial}{\partial y}$. (15 marks) (b) The velocity of a train which starts from rest is given by the following table, the time being reckoned in minutes from the start and the velocity in km/hour: | $t$ (minutes) | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | |---|---|---|---|---|---|---|---|---|---|---| | $v$ (km/hour) | 16 | 28.8 | 40 | 46.4 | 51.2 | 32 | 17.6 | 8 | 3.2 | 0 | Using Simpson's $\frac{1}{3}$rd rule, estimate approximately in km the total distance run in 20 minutes. (15 marks) (c) Two point vortices each of strength $k$ are situated at $(\pm a, 0)$ and a point vortex of strength $-\frac{k}{2}$ is situated at the origin. Show that the fluid motion is stationary and also find the equations of streamlines. If the streamlines, which pass through the stagnation points, meet the $x$-axis at $(\pm b, 0)$, then show that $3\sqrt{3}(b^2-a^2)^2 = 16a^3b$. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) आंशिक अवकल समीकरण $$(D^2 + DD' - 6D'^2)z = x^2 \sin(x+y)$$ जहाँ $D \equiv \frac{\partial}{\partial x}$ तथा $D' \equiv \frac{\partial}{\partial y}$, का व्यापक हल ज्ञात कीजिये। (15 अंक) (b) एक रेलगाड़ी, जो कि विश्राम से चलना प्रारंभ करती है, का वेग निम्नलिखित सारणी द्वारा दिया गया है। प्रस्थान से समय की गणना मिनट में तथा वेग की कि० मी०/घंटा में की गयी है: | $t$ (मिनट) | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | |---|---|---|---|---|---|---|---|---|---|---| | $v$ (कि० मी०/घंटा) | 16 | 28.8 | 40 | 46.4 | 51.2 | 32 | 17.6 | 8 | 3.2 | 0 | सिम्पसन के $\frac{1}{3}$ नियम का उपयोग करके 20 मिनट में तय की गयी कुल दूरी (लगभग) का आकलन कि० मी० में कीजिये। (15 अंक) (c) दो बिंदु भ्रमिल, जहाँ प्रत्येक का सामर्थ्य $k$ है, $(\pm a, 0)$ पर स्थित हैं तथा $-\frac{k}{2}$ सामर्थ्य का एक बिंदु भ्रमिल, मूलबिंदु पर स्थित है। दर्शाइये कि तरल गति अचल है तथा धारारेखाओं के समीकरण भी ज्ञात कीजिये। यदि धारारेखाएँ, जो कि प्रगतिरोध बिंदुओं (स्टैगनेशन पॉइंट) से गुजरती हैं, $x$-अक्ष पर $(\pm b, 0)$ पर मिलती हैं, तब दर्शाइये कि $3\sqrt{3}(b^2-a^2)^2 = 16a^3b$। (20 अंक)
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How this answer will be evaluated
Approach
Solve this three-part numerical problem by allocating approximately 30% time to part (a) PDE solution, 30% to part (b) numerical integration, and 40% to part (c) vortex dynamics. Begin with the complementary function and particular integral for (a), apply Simpson's 1/3rd rule with proper unit conversion for (b), and establish complex potential, velocity field, and streamline equations for (c). Conclude each part with verified final answers and proper dimensional analysis.
Key points expected
- Part (a): Factorize the operator (D² + DD' - 6D'²) = (D + 3D')(D - 2D'), find complementary function φ₁(y - 3x) + φ₂(y + 2x), and use correct particular integral method for x²sin(x+y)
- Part (b): Convert time interval to hours (h = 2/60 = 1/30 hour), verify even number of intervals for Simpson's 1/3rd rule, and compute distance = ∫v dt with proper unit handling
- Part (c): Construct complex potential w = -ik/2π[ln(z-a) + ln(z+a) - ½ln(z)], show dw/dz = 0 at stagnation points, derive streamline equations ψ = constant
- Part (c): Locate stagnation points on x-axis by solving velocity equations, find streamline passing through them, and verify the given algebraic relation 3√3(b²-a²)² = 16a³b
- Correct handling of units throughout: km/hour to km for distance, consistent time conversions, and dimensionless verification in part (c)
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly factorizes the PDE operator in (a), converts units properly in (b) recognizing h=1/30 hour, and sets up complex potential with image vortices in (c); identifies all boundary conditions and constraints | Partially correct factorization or unit conversion with minor errors; complex potential missing one term or incorrect sign convention | Wrong operator factorization, fails to convert minutes to hours, or completely incorrect potential setup; misses essential boundary conditions |
| Method choice | 20% | 10 | Uses standard particular integral formula for x²sin(x+y) in (a), applies Simpson's 1/3rd rule correctly with proper weight pattern 1-4-2-4-...-1 in (b), and employs complex variable methods for vortex analysis in (c) | Correct general approach but substitutes wrong formula or uses trapezoidal rule instead of Simpson's; partial complex analysis | Wrong method entirely: variation of parameters for PDE, Newton-Cotes instead of Simpson's, or purely real analysis for vortices |
| Computation accuracy | 20% | 10 | Accurate particular integral with correct polynomial coefficients in (a), precise Simpson's calculation yielding distance ≈ 10.67 km in (b), and exact stagnation point locations with verified algebraic identity in (c) | Minor arithmetic slips in polynomial coefficients or Simpson's summation; correct stagnation points but algebraic verification incomplete | Major computational errors: wrong coefficients, incorrect Simpson's weights, or failure to solve quartic for stagnation points; algebraic identity not attempted |
| Step justification | 20% | 10 | Shows each step: operator factorization, trial function selection for PI, Simpson's coefficient pattern justification, and derivation of velocity from complex potential; explains why motion is stationary in (c) | Some steps shown but gaps in reasoning; assumes results without derivation or skips verification of stationary motion | No intermediate steps shown; jumps to answers without derivation; no explanation of physical significance |
| Final answer & units | 20% | 10 | Complete general solution with arbitrary functions in (a), distance in km with appropriate precision in (b), explicit streamline equations and verified identity in (c); all units consistent and stated | Correct answers but units missing or inconsistent; streamline equations implicit or verification incomplete | Answers without units, missing arbitrary functions, or final algebraic identity not established; wrong order of magnitude for distance |
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