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GATE 2021 MA – Question 38

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

Solve xu_x+u_y=1 with u(x,0)=−x²,x>0. Find u(2,1).

  1. 1−e^(−2)
  2. 1+4e^(−2)
  3. 1−4e^(−2)
  4. 1+2e^(−2)

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Correct answer: (C) 1−4e^(−2)

Explanation

Using y as the characteristic parameter, dx/dy=x gives invariant xe⁻ʸ and du/dy=1 gives u−y constant. Therefore u(x,y)=y−(xe⁻ʸ)². At (2,1), this is **1−4e^(−2), C**.

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