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GATE 2022 MA – Question 62

Partial Differential Equations · First Order Equations · 2 marks · Numerical answer

For xu_x+(x²+y)u_y=u on ℝ², u(2,y)=y−4. Find u(1,2).

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Correct answer: 1 to 1

Explanation

Along characteristics dy/dx=x+y/x, so d(y/x)/dx=1 and c=y/x−x is constant. Also du/dx=u/x, so u/x is constant along each characteristic and u=xF(c). At x=2, c=(y−4)/2 and u=y−4=2c, giving F(c)=c. Hence u=y−x², and **u(1,2)=1**.

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