Paper II — Q7
(a) Find the general solution of the partial differential equation (D² - D'² - 3D + 3D')z = xy + e^(x+2y) where D ≡ ∂/∂x and D' ≡…
Find the general solution of the partial differential equation (D² - D'² - 3D + 3D')z = xy + e^(x+2y) where D ≡ ∂/∂x and D' ≡ ∂/∂y. 15 marks
Solve the system of equations 3x₁ + 9x₂ - 2x₃ = 11 4x₁ + 2x₂ + 13x₃ = 24 4x₁ - 2x₂ + x₃ = -8 correct up to 4 significant figures by using Gauss-Seidel method after verifying whether the method is applicable in your transformed form of the system. 15 marks
Show that q⃗ = λ(-yî + xĵ)/(x² + y²), (λ = constant) is a possible incompressible fluid motion. Determine the streamlines. Is the kind of the motion potential? If yes, then find the velocity potential. 20 marks
हिंदी में प्रश्न पढ़ें
आंशिक अवकल समीकरण (D² - D'² - 3D + 3D')z = xy + e^(x+2y) का व्यापक हल प्राप्त कीजिए, जहाँ D ≡ ∂/∂x तथा D' ≡ ∂/∂y है। 15 अंक
समीकरणों के निकाय 3x₁ + 9x₂ - 2x₃ = 11 4x₁ + 2x₂ + 13x₃ = 24 4x₁ - 2x₂ + x₃ = -8 का गॉस-सीडल विधि द्वारा 4 सार्थक अंकों तक सही हल प्राप्त कीजिए, यह सत्यापन करने के बाद कि क्या यह विधि आपके द्वारा निकाय के रूपांतरित रूप में अनुप्रयोज्य है। 15 अंक
दर्शाइए कि q⃗ = λ(-yî + xĵ)/(x² + y²), (λ = स्थिरांक) एक संभाव्य असंपीड्य तरल गति है। धारा-रेखाएँ निकालिए। क्या गति का प्रकार विभव है? यदि हाँ, तो वेग विभव निकालिए। 20 अंक
Model answer
Drafted with AI by UPSC Answer Check against this question's marking rubric, to the expected length. UPSC does not publish answers for Mains — this is one way to score well, not an official key. How we build rubrics and answers.
(a) General solution of the PDE
The operator factorises: D² − D′² − 3D + 3D′ = (D − D′)(D + D′ − 3).
Complementary function: (D − D′)z = 0 gives φ(x + y), and (D + D′ − 3)z = 0 gives e^(3x)ψ(y − x). So z_c = φ(x + y) + e^(3x)ψ(y − x).
For e^(x+2y), the operator value at D = 1, D′ = 2 is 1 − 4 − 3 + 6 = 0, so try z₁ = (Ax + B)e^(x+2y). This gives Lz₁ = −A e^(x+2y), so A = −1 and z₁ = −x e^(x+2y).
For xy, put w = (D − D′)z₂ and solve (D + D′ − 3)w = xy with w = Axy + Bx + Cy + E. Comparing coefficients: A = −1/3, B = C = −1/9, E = −2/27. Then (D − D′)z₂ = w is satisfied by
z₂ = xy²/6 + y³/18 + xy/9 + y²/9 − 2x/27.
General solution: z = φ(x + y) + e^(3x)ψ(y − x) − x e^(x+2y) + xy²/6 + y³/18 + xy/9 + y²/9 − 2x/27.
(b) Gauss–Seidel solution
As given, the system is not diagonally dominant (row 1 has 3 < 9 + 2). Reorder the equations as
4x₁ − 2x₂ + x₃ = −8, 3x₁ + 9x₂ − 2x₃ = 11, 4x₁ + 2x₂ + 13x₃ = 24.
Now 4 > 3, 9 > 5, 13 > 6, so the matrix is strictly diagonally dominant and Gauss–Seidel converges.
Iteration formulae, using the latest values: x₁ = (−8 + 2x₂ − x₃)/4, x₂ = (11 − 3x₁ + 2x₃)/9, x₃ = (24 − 4x₁ − 2x₂)/13.
Starting from (0, 0, 0): sweep 1 gives (−2.000, 1.889, 2.171); sweep 2 gives (−1.598, 2.237, 1.994); sweep 3 gives (−1.380, 2.125, 1.944); sweep 4 gives (−1.423, 2.129, 1.957); sweep 6 gives (−1.423, 2.131, 1.956); sweep 7 repeats these four-figure values.
Solution: x₁ = −1.423, x₂ = 2.131, x₃ = 1.956.
(c) Vortex flow
Let u = −λy/r², v = λx/r², r² = x² + y².
Incompressible: ∂u/∂x = 2λxy/r⁴ and ∂v/∂y = −2λxy/r⁴, so ∇·q = 0 for r ≠ 0. This is a possible incompressible motion everywhere except the origin.
Streamlines: dy/dx = v/u = −x/y gives x dx + y dy = 0, so x² + y² = constant. The streamlines are concentric circles about the origin; the stream function is ψ = −λ ln r.
Potential motion: ∂v/∂x = λ(y² − x²)/r⁴ = ∂u/∂y, so the curl is zero for r ≠ 0 and the motion is irrotational. Hence a velocity potential exists. Since ∂/∂x[λ tan⁻¹(y/x)] = −λy/r² = u and ∂/∂y[λ tan⁻¹(y/x)] = λx/r² = v,
Φ = λ tan⁻¹(y/x) + constant = λθ + constant.
Φ is multi-valued: the circulation round the origin is 2πλ, so Φ is single-valued only in a region that does not enclose the origin.
How this answer will be evaluated
Framework: UPSC Mathematics Paper 2
What earns full marks: Rigorous derivation with all steps, correct factorization, and verified convergence.
Evaluation rubric
Each sub-part is marked on its own, against the marks and word limit printed on the paper.
- (a) Find the general solution of the PDE (D² − D′² − 3D + 3D′)z = xy + e^(x+2y) 15 marks
derive— given → assumptions → stepwise derivation → result → check
Must cover
- Factorisation (D − D′)(D + D′ − 3) and complementary function φ(x + y) + e^(3x)ψ(y − x)
- Resonant particular integral −x e^(x+2y)
- Particular integral for xy and the complete general solution
Loses marks
- Using 1/(1 − 4 − 3 + 6) without noticing it is zero
- Omitting arbitrary functions from the complementary function
- Wrong factorisation of the operator
Earns more
- Correct treatment of the resonance for e^(x+2y)
- Particular integral for xy by a two-step or undetermined-coefficient method
- Final general solution assembled correctly
Extra mark
- Verifies the particular integrals by substitution
- (b) Solve the system by Gauss–Seidel to four significant figures after checking applicability 15 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- Equations reordered to a strictly diagonally dominant form
- Iteration formulae using the latest values
- x₁ = −1.423, x₂ = 2.131, x₃ = 1.956
Loses marks
- Applying Gauss–Seidel without checking dominance
- Using old values within a sweep
- Stopping before four-figure stability
Earns more
- Dominance checked row by row
- At least four iterations shown correctly
- Convergence to four significant figures indicated
Extra mark
- Cross-checks the answer by substituting in the original equations
- (c) Show the flow is incompressible, find the streamlines and test for a velocity potential 20 marks
derive— given → assumptions → stepwise derivation → result → check
Must cover
- ∇·q = 0 shown for r ≠ 0
- Streamlines x² + y² = constant
- Irrotational, with Φ = λ tan⁻¹(y/x)
Loses marks
- Claiming the flow is irrotational at the origin
- Wrong streamline family
- Omitting the condition r ≠ 0
Earns more
- Partial derivatives computed correctly
- Streamlines obtained from dy/dx = v/u
- Curl shown to vanish and potential verified
Extra mark
- Notes the multi-valued potential and circulation 2πλ
What "Solve" is asking you to do
Choose the method, then carry it through to a final answer. Identifying what kind of problem this is and why that method applies is the first thing marked; a correct figure arrived at invisibly earns almost nothing.
Structure that answers it
Given data and what is required → method chosen, with the reason it applies → set-up (equation, circuit, free body, trial balance) → working, step by step → answer with units and any condition of validity
Where marks are lost
Doing the middle steps mentally and writing only the result. In mathematics papers, a further loss comes from giving a decimal where the exact value in surds or fractions was wanted, or from skipping the justification a part explicitly asks for.
Practice this exact question
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