Q8
(a) Solve the following initial value problem by using Laplace transform technique : $$\frac{d^2y}{dt^2} - 4\frac{dy}{dt} + 3y(t) = f(t),$$ $y(0) = 1$, $y'(0) = 0$ and $f(t)$ is a given function of $t$. 15 (b) A particle is projected from an apse at a distance $\sqrt{c}$ from the centre of force with a velocity $\sqrt{\frac{2\lambda}{3}c^3}$ and is moving with central acceleration $\lambda(r^5 - c^2r)$. Find the path of motion of this particle. Will that be the curve $x^4 + y^4 = c^2$ ? 20 (c) For a scalar point function $\phi$ and vector point function $\vec{f}$, prove the identity $\nabla \cdot (\phi\vec{f}) = \nabla\phi \cdot \vec{f} + \phi(\nabla \cdot \vec{f})$. Also find the value of $\nabla \cdot \left(\frac{f(r)}{r}\vec{r}\right)$ and then verify stated identity. 15
हिंदी में प्रश्न पढ़ें
(a) लाप्लास रूपांतर प्रविधि का उपयोग कर निम्नलिखित प्रारंभिक मान समस्या को हल कीजिए। $$\frac{d^2y}{dt^2} - 4\frac{dy}{dt} + 3y(t) = f(t),$$ $y(0) = 1$, $y'(0) = 0$ और $f(t)$, $t$ का एक दिया गया फलन है। 15 (b) एक कण, बल-केंद्र से $\sqrt{c}$ दूरी पर स्थित एक स्तब्धिका से $\sqrt{\frac{2\lambda}{3}c^3}$ वेग से प्रक्षेपित किया जाता है और यह केंद्रीय त्वरण $\lambda(r^5 - c^2r)$ से गतिशील है। इस कण की गति का पथ ज्ञात कीजिए। क्या यह वक्र $x^4 + y^4 = c^2$ होगा ? 20 (c) एक अदिश बिंदु फलन $\phi$ और सदिश बिंदु फलन $\vec{f}$ के लिये निम्नलिखित सर्वसमिका सिद्ध कीजिए $$\nabla \cdot (\phi\vec{f}) = \nabla\phi \cdot \vec{f} + \phi(\nabla \cdot \vec{f})$$ $$\nabla \cdot \left(\frac{f(r)}{r}\vec{r}\right)$$ का मान भी ज्ञात कीजिए और तब उल्लेखित सर्वसमिका का सत्यापन कीजिए। 15
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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) Laplace transform IVP, 40% to part (b) central force motion and orbit determination, and 30% to part (c) vector calculus identity proof and verification. Begin with clear statement of given conditions, apply appropriate mathematical techniques systematically, show all computational steps explicitly, and conclude with precise final answers for each sub-part.
Key points expected
- Part (a): Correct application of Laplace transform to second-order ODE with proper handling of initial conditions y(0)=1, y'(0)=0 and symbolic f(t)
- Part (a): Accurate partial fraction decomposition and inverse Laplace transform to obtain complete solution y(t)
- Part (b): Correct formulation of central force problem using Binet's equation or energy equation, identifying the given acceleration λ(r⁵ - c²r)
- Part (b): Derivation of orbital equation and verification whether the path satisfies x⁴ + y⁴ = c² through polar to Cartesian conversion
- Part (c): Rigorous proof of vector identity ∇·(φf⃗) = ∇φ·f⃗ + φ(∇·f⃗) using component-wise expansion or index notation
- Part (c): Correct computation of ∇·(f(r)/r · r⃗) using spherical/polar coordinate formulas and explicit verification of the identity
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly writes ℒ{y''}, ℒ{y'}, applies initial conditions; for (b): sets up central force equations with proper energy/angular momentum; for (c): defines φ, f⃗, r⃗ clearly with coordinate system specified | Minor errors in initial setup such as sign errors in Laplace transform of derivatives or incorrect identification of force components in central motion | Fundamental errors like treating y(0) and y'(0) interchangeably, confusing central acceleration with potential, or missing coordinate specifications |
| Method choice | 20% | 10 | For (a): uses convolution theorem appropriately for general f(t); for (b): selects Binet's equation or energy method suited to r⁵ - c²r form; for (c): employs systematic component expansion or tensor notation | Correct but inefficient methods, such as expanding all components unnecessarily or using energy method when Binet's is cleaner | Inappropriate methods like attempting series solutions for Laplace transform problem or using Cartesian coordinates throughout central force problem |
| Computation accuracy | 20% | 10 | For (a): accurate partial fractions (s²-4s+3)=(s-1)(s-3) and clean inverse transforms; for (b): correct integration of orbital equation yielding explicit r(θ); for (c): error-free differentiation of f(r)/r · r⃗ | Computational slips like arithmetic errors in partial fraction coefficients or sign errors in divergence calculation that propagate partially | Major computational failures such as incorrect factorization of characteristic equation, wrong integration of r⁵ term, or fundamental errors in gradient/divergence formulas |
| Step justification | 20% | 10 | Each mathematical step explicitly justified: why ℒ⁻¹{1/(s-1)(s-3)} uses partial fractions, physical meaning of apsidal distance in orbit, and logical flow from ∇·(φf⃗) expansion to final verification | Steps shown but with gaps in reasoning, such as stating results without showing intermediate algebra or assuming standard formulas without citation | Unjustified leaps, missing steps in derivation, or 'it can be shown that' gaps where critical reasoning should appear |
| Final answer & units | 20% | 10 | For (a): complete solution y(t) expressed with convolution integral for arbitrary f(t); for (b): explicit orbital equation and clear yes/no on x⁴+y⁴=c² with justification; for (c): simplified divergence expression and verified identity | Correct final forms but incompletely simplified, or missing explicit verification in part (c), or unclear conclusion on curve identification | Missing final answers, incorrect boxed conclusions, or failure to address whether x⁴+y⁴=c² is indeed the path |
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