Q5
(a) Show that the general solution of the differential equation $\frac{dy}{dx} + Py = Q$ can be written in the form $y = \frac{Q}{P} - e^{-\int P \, dx}\left\{C + \int e^{\int P \, dx} \, d\left(\frac{Q}{P}\right)\right\}$, where P, Q are non-zero functions of x and C, an arbitrary constant. 10 (b) Show that the orthogonal trajectories of the system of parabolas : $x^2 = 4a(y + a)$ belong to the same system. 10 (c) A body of weight w rests on a rough inclined plane of inclination $\theta$, the coefficient of friction, $\mu$, being greater than tan $\theta$. Find the work done in slowly dragging the body a distance 'b' up the plane and then dragging it back to the starting point, the applied force being in each case parallel to the plane. 10 (d) A projectile is fired from a point O with velocity $\sqrt{2gh}$ and hits a tangent at the point P(x, y) in the plane, the axes OX and OY being horizontal and vertically downward lines through the point O, respectively. Show that if the two possible directions of projection be at right angles, then $x^2 = 2hy$ and then one of the possible directions of projection bisects the angle POX. 10 (e) Show that $\vec{A} = (6xy + z^3)\hat{i} + (3x^2 - z)\hat{j} + (3xz^2 - y)\hat{k}$ is irrotational. Also find $\phi$ such that $\vec{A} = \nabla\phi$. 10
हिंदी में प्रश्न पढ़ें
(a) दर्शाइए कि अवकल समीकरण $\frac{dy}{dx} + Py = Q$ का व्यापक हल $$y = \frac{Q}{P} - e^{-\int P \, dx}\left\{C + \int e^{\int P \, dx} \, d\left(\frac{Q}{P}\right)\right\}$$ के रूप में लिखा जा सकता है, जहाँ P, Q, x के शून्येतर फलन हैं तथा C एक स्वेच्छ अचर है। 10 (b) दर्शाइए कि परवलयों के निकाय : $x^2 = 4a(y + a)$ के लंबकोणीय संहेडी, उसी निकाय में स्थित होते हैं। 10 (c) w भार का एक पिंड, $\theta$ कोण से झुके हुए एक रूक्ष समतल पर स्थित है, घर्षण गुणांक $\mu$, tan $\theta$ से अधिक है। पिंड को समतल पर ऊपर की तरफ 'b' दूरी तक धीरे-धीरे खींचने तथा वापस आरंभिक बिंदु तक खींचने में किए गए कार्य को ज्ञात कीजिए, जहाँ लगाया गया बल प्रत्येक दशा में समतल के समांतर है। 10 (d) एक प्रक्षेप्य $\sqrt{2gh}$ वेग के साथ बिंदु O से प्रक्षेपित किया गया तथा समतल के बिंदु P(x, y) पर स्पर्शरेखा से टकराता है जहाँ अक्ष OX तथा OY क्रमशः बिंदु O से क्षैतिज तथा अधोमुखी उर्ध्वाधर रेखाएँ हैं। यदि प्रक्षेपण की दो संभव दिशाएँ समकोण पर हों, तो दर्शाइए कि $x^2 = 2hy$ तथा प्रक्षेपण की संभव दिशाओं में से एक, कोण POX को द्विभाजित करती है। 10 (e) दर्शाइए कि $\vec{A} = (6xy + z^3)\hat{i} + (3x^2 - z)\hat{j} + (3xz^2 - y)\hat{k}$ अघूर्णी है। $\phi$ को भी ज्ञात कीजिए जबकि $\vec{A} = \nabla\phi$। 10
Directive word: Prove
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How this answer will be evaluated
Approach
Prove the required results across all five sub-parts with rigorous mathematical derivations. For (a), derive the alternative form of the general solution using the standard integrating factor method; for (b), find the differential equation of the family and its orthogonal trajectories; for (c), analyze forces and calculate work done in both directions; for (d), use projectile equations and the condition of perpendicular directions; for (e), compute curl and integrate to find the scalar potential. Allocate approximately 20% time to each part given equal marks distribution.
Key points expected
- Part (a): Derivation of the alternative form using integrating factor e^(∫Pdx), recognition that d(Q/P)/dx expansion leads to the required structure, and proper handling of the constant of integration
- Part (b): Formation of differential equation by eliminating parameter 'a', obtaining dy/dx = x/(2a), substitution to get orthogonal trajectories differential equation, and verification that the resulting equation represents the same family of parabolas
- Part (c): Correct force analysis showing applied force = w(sinθ + μcosθ) upward and w(μcosθ - sinθ) downward, integration to find work done as wb(sinθ + μcosθ) + wb(μcosθ - sinθ) = 2wbμcosθ
- Part (d): Use of trajectory equation y = xtanα - (gx²sec²α)/(4gh), condition that product of slopes m₁m₂ = -1 for perpendicular directions leading to x² = 2hy, and verification that one bisector condition holds
- Part (e): Calculation of curl A showing all components vanish (∇×A = 0), and integration to find φ = 3x²y + xz³ - yz + C with correct verification that ∇φ = A
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies integrating factor for (a), properly eliminates parameter for orthogonal trajectories in (b), draws correct free-body diagrams with all forces for (c), sets up correct projectile equation with given velocity for (d), and correctly computes curl components for (e) | Minor errors in setup such as wrong integrating factor form, incomplete elimination of parameter, missing one force component, or one incorrect curl component | Fundamental errors like treating P,Q as constants, failing to eliminate parameter, major force analysis mistakes, wrong trajectory equation, or multiple curl errors |
| Method choice | 20% | 10 | Uses standard integrating factor method with clever rearrangement for (a), standard orthogonal trajectory technique for (b), work-energy or force-integration approach for (c), condition on roots of quadratic in tanα for (d), and exact differential/integration by parts for (e) | Correct general methods but inefficient approaches or missing optimal techniques like not recognizing the exact differential form in (a) or (e) | Wrong methods such as variation of parameters for (a), wrong trajectory family approach for (b), energy methods ignoring friction for (c), or scalar/vector potential confusion for (e) |
| Computation accuracy | 20% | 10 | Flawless algebraic manipulation, correct integration of d(Q/P), accurate orthogonal trajectory derivation, precise work calculation yielding 2wbμcosθ, correct quadratic condition application, and exact potential function with all cross-terms verified | Minor computational slips like sign errors in integration, arithmetic errors in final simplification, or small errors in potential function coefficients | Major computational errors leading to wrong final forms, incorrect integration results, wrong work formula, failure to establish x²=2hy, or incorrect potential function |
| Step justification | 20% | 10 | Clear justification for each transformation in (a), explicit reasoning for orthogonal condition in (b), physical reasoning for 'slowly dragged' implying equilibrium in (c), clear derivation of perpendicularity condition via sum and product of roots in (d), and verification that mixed partials match for (e) | Some steps justified but gaps in reasoning, missing explanation for key transformations, or insufficient physical justification | Missing crucial justifications, unexplained leaps in algebra, no physical reasoning for equilibrium assumption, or no verification of potential function |
| Final answer & units | 20% | 10 | All five parts yield exactly the required forms: stated solution structure for (a), demonstrated same family membership for (b), work = 2wbμcosθ with proper units for (c), established x²=2hy and angle bisection for (d), and φ = 3x²y + xz³ - yz + C for (e) | Correct final answers for most parts but one incorrect or incomplete, or missing arbitrary constant in potential function | Multiple incorrect final answers, wrong forms, missing crucial components like the angle bisection proof, or completely wrong potential function |
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