GATE CE 2026 Set 2 — Question 30
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Engineering Mathematics → Partial Differential Equations → 1D Wave Equation (Second-Order)
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Question
A partial differential equation is given below.
∂x2∂2u−∂y2∂2u=0Possible solution(s) is/are:
Correct answer
(A) (x + y)⁵; (C) cos(x + y)
Solution
The given equation is the one-dimensional wave equation:
∂x2∂2u−∂y2∂2u=0The general solution is of the form
u(x,y)=f(x+y)+g(x−y).
- Option (A): u=(x+y)5. Here uxx=20(x+y)3 and uyy=20(x+y)3. Thus uxx−uyy=0. Correct.
- Option (B): u=(x−2y)3. Here uxx=6(x−2y) and uyy=24(x−2y). Thus uxx−uyy=0. Incorrect.
- Option (C): u=cos(x+y). Here uxx=−cos(x+y) and uyy=−cos(x+y). Thus uxx−uyy=0. Correct.
- Option (D): u=sin(x−2y). Here uxx=−sin(x−2y) and uyy=−4sin(x−2y). Thus uxx−uyy=0. Incorrect.
Therefore, (A) and (C) are valid solutions.
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