Mathematics · Partial Differential Equations
Let u : ℝ 2 → ℝ be the solution to the Cauchy problem: ∂ x u + 2 ∂ y u = 0 for (x, y) ∈ ℝ 2 , and u(x, y) = sin(x) for y = 3x + 1, x ∈ ℝ. Let v : ℝ 2 → ℝ be the solution to the Cauchy problem: ∂ x v + 2 ∂ y v = 0 for (x, y) ∈ ℝ 2 , and v(x, 0) = sin(x) for x ∈ ℝ. Let Ω = [0, 1] × [0, 1]. Which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1v vanishes along a line in Ω.
- 2v changes sign in the interior of Ω.✓
- 3u(x, y) = v(x, y) along a line in Ω.
- 4u changes sign in the interior of Ω.
Correct answer
v changes sign in the interior of Ω.
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