Q8
(a) (i) Find the general and singular solutions of the differential equation : $(x^2 - a^2)p^2 - 2xyp + y^2 + a^2 = 0$, where $p = \frac{dy}{dx}$. Also give the geometric relation between the general and singular solutions. 10 (ii) Solve the following differential equation : $$(3x + 2)^2\frac{d^2y}{dx^2} + 5(3x + 2)\frac{dy}{dx} - 3y = x^2 + x + 1$$ 10 (b) A chain of n equal uniform rods is smoothly jointed together and suspended from its one end A₁. A horizontal force $\vec{P}$ is applied to the other end Aₙ₊₁ of the chain. Find the inclinations of the rods to the downward vertical line in the equilibrium configuration. 15 (c) Using Gauss' divergence theorem, evaluate $\iint\limits_{S} \vec{F}.\vec{n}$ dS, where $\vec{F} = x\hat{i} - y\hat{j} + (z^2-1)\hat{k}$ and S is the cylinder formed by the surfaces z = 0, z = 1, x² + y² = 4. 15
हिंदी में प्रश्न पढ़ें
(a) (i) अवकल समीकरण : $(x^2 - a^2)p^2 - 2xyp + y^2 + a^2 = 0$, जहाँ $p = \frac{dy}{dx}$, के व्यापक व विचित्र हलों को ज्ञात कीजिए । व्यापक व विचित्र हलों के बीच ज्यामितीय संबंध को भी दीजिए । 10 (ii) निम्नलिखित अवकल समीकरण को हल कीजिए : $$(3x + 2)^2\frac{d^2y}{dx^2} + 5(3x + 2)\frac{dy}{dx} - 3y = x^2 + x + 1$$ 10 (b) n बराबर एकसमान छड़ों की एक श्रृंखला एक-दूसरे के साथ चिकने रूप से जुड़ी हुई है तथा इसके एक सिरे A₁ से लटकी हुई है । एक क्षैतिज बल $\vec{P}$ श्रृंखला के दूसरे सिरे Aₙ₊₁ पर लगाया गया है । साम्य विन्यास में अधोमुखी उद्वाधर रेखा से छड़ों के झुकाव ज्ञात कीजिए । 15 (c) गॉस के अपसरण प्रमेय का उपयोग करके $\iint\limits_{S} \vec{F}.\vec{n}$ dS का मान निकालिए, जहाँ $\vec{F} = x\hat{i} - y\hat{j} + (z^2-1)\hat{k}$ तथा S, पृष्ठों z = 0, z = 1, x² + y² = 4 द्वारा बना हुआ बेलन है । 15
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How this answer will be evaluated
Approach
Solve each sub-part systematically: spend ~40% time on (a) covering both differential equations (Clairaut's form recognition and Euler-Cauchy transformation), ~30% on (b) for the static equilibrium of linked rods using virtual work or force balance, and ~30% on (c) for applying divergence theorem with careful handling of cylindrical surface and end caps. Present solutions with clear identification of method, step-by-step working, and boxed final answers.
Key points expected
- (a)(i) Recognize the equation as Clairaut's form, extract general solution y = px + f(p), find singular solution by eliminating p, and state that singular solution is the envelope of general solution family
- (a)(ii) Apply substitution 3x+2 = e^t to convert to constant coefficient linear ODE, solve homogeneous part, find particular integral for quadratic RHS, and back-substitute
- (b) Set up equilibrium conditions for n-rod chain using tension propagation, horizontal/vertical force balance at each joint, derive recurrence for inclinations θ_k, and obtain closed form tan θ_k = P/((2k-1)W/2) where W is rod weight
- (c) Compute div F = 1 - 1 + 2z = 2z, evaluate volume integral ∫∫∫ 2z dV over cylinder 0≤z≤1, x²+y²≤4, and verify by direct surface integration on three surfaces (curved wall, top disk, bottom disk)
- Geometric interpretation for (a)(i): singular solution touches each member of general solution family (parabola envelope)
- For (b), recognize the pattern forms arithmetically progressing horizontal reactions and express final inclination angles in terms of P, W, and n
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies (a)(i) as Clairaut's equation y = px ± a√(p²+1), (a)(ii) as Euler-Cauchy type requiring substitution, (b) as static equilibrium with proper free-body diagrams, and (c) as divergence theorem application with cylinder geometry properly segmented | Identifies most equation types but misses subtlety (e.g., recognizes Clairaut but writes wrong form, or attempts divergence theorem but confuses surface orientation) | Misidentifies equation types (treats (a)(i) as quadratic in p without Clairaut insight, or attempts (a)(ii) with wrong substitution), or sets up wrong geometry for (c) |
| Method choice | 20% | 10 | Uses optimal methods: Clairaut standard procedure, Euler-Cauchy substitution 3x+2=e^t, virtual work or tension method for (b), and divergence theorem with cylindrical coordinates for (c); no redundant calculations | Uses workable but suboptimal methods (e.g., treats (a)(ii) by variation of parameters directly without simplifying substitution, or computes (c) by direct surface integration without divergence theorem) | Uses inappropriate methods (e.g., series solution for (a)(ii), or attempts Lagrange multipliers for (b)) |
| Computation accuracy | 20% | 10 | Flawless algebra: correct elimination of p for singular solution, accurate characteristic roots and particular integral in (a)(ii), correct recurrence solution tan θ_k = P/((n-k+½)W) for (b), and exact evaluation 4π for (c) | Minor computational slips (sign errors in characteristic equation, arithmetic errors in recurrence coefficients, or volume integral bounds error) that don't completely derail solution | Major computational errors (wrong discriminant in (a)(i), incorrect PI form in (a)(ii), fundamental error in equilibrium equations for (b), or div F calculation error in (c)) |
| Step justification | 20% | 10 | Every non-trivial step justified: why p-elimination gives envelope, why substitution works for (a)(ii), physical reasoning for tension directions in (b), and explicit verification that divergence theorem applies (continuously differentiable F, closed surface) | Some steps justified but gaps remain (e.g., states singular solution without envelope proof, or applies divergence theorem without checking conditions) | Minimal justification—jumps between steps without explanation, or presents unmotivated formulas |
| Final answer & units | 20% | 10 | All four answers clearly presented: (a)(i) general solution y = cx ± a√(c²+1), singular solution x²+y²=a² with envelope statement; (a)(ii) complete y(x); (b) explicit θ_k formula; (c) numerical value 4π with verification of zero flux through end caps | Most answers present but incomplete (missing singular solution, or (b) answer in implicit form, or (c) without verification) | Missing or wrong final answers, or answers without geometric/physical interpretation where requested |
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