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GATE 2025 MA – Question 51

Partial Differential Equations · First Order Equations · 2 marks · Multiple select

On ℝ,t>0, u_t+3u_x=u and v_t+3v_x=v², both with initial data cosx. Which statements hold?

  1. |u(x,t)|≤eᵗ everywhere.
  2. v(x,1) is undefined for certain x.
  3. v(x,1) is undefined for every x.
  4. u(2π,π)=−e^π

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Show answer and explanation

Correct answer: (A) |u(x,t)|≤eᵗ everywhere.; (B) v(x,1) is undefined for certain x.; (D) u(2π,π)=−e^π

Explanation

Along characteristics x=ξ+3t, $u=e^t\cos\xi$ and $v=\cos\xi/[1-t\cos\xi]$ until blow-up. Hence A holds. At t=1, denominator vanishes at ξ=2πk, so x=3+2πk, proving B but not C. At (2π,π), ξ=−π, so u=−e^π, proving D. **A,B,D**.

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