GATE CE 2025 Set 1 — Question 28
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Engineering Mathematics → Partial Differential Equations
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Question
Which of the following equations belong/belongs to the class of second-order, linear, homogeneous partial differential equations:
Correct answer
(B) (∂² u)/(∂ x²) + (∂² u)/(∂ y²) + (∂² u)/(∂ z²) = 0
Solution
A partial differential equation is second-order if the highest derivative is of order 2. It is linear if the dependent variable and its derivatives appear only to the first power and are not multiplied together. It is homogeneous if every term contains the dependent variable or its derivatives.(A) : Second-order, linear, but non-homogeneous due to the term .
(B) : Second-order, linear, and homogeneous (Laplace equation).
(C) \frac{\partial u}{\partial t} = c \frac{\partial u \partial x}: First-order.
(D) : Second-order, but non-linear because the second derivative is squared.
(B) : Second-order, linear, and homogeneous (Laplace equation).
(C) \frac{\partial u}{\partial t} = c \frac{\partial u \partial x}: First-order.
(D) : Second-order, but non-linear because the second derivative is squared.
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