Question

Which of the following are not the solution of \( \frac{\partial^2 z}{\partial x^2} = \frac{\partial^2 z}{\partial y^2} \)? A. \(z = f(y+x) + f(y-x)\)
B. \(z = f(y+x) + f(y-x)\)
C. \(z = f(x^2 - y^2)\)
D. \(z = f(x^2 + y^2)\)

• A. A, B and C only
• B. B, C and D only
• C. A, C and D only
• D. A, B and D only

Hint:

Wave-type PDE → solution depends on \(x+y\) and \(x-y\).

Answer 1

The Correct Option is B

Concept: Given PDE: \[ \frac{\partial^2 z}{\partial x^2} = \frac{\partial^2 z}{\partial y^2} \] General solution: \[ z = f(x+y) + g(x-y) \]

Step 1: Check option A and B.
They match general solution form → valid solutions.

Step 2: Check option C.
\[ z = f(x^2 - y^2) \] Differentiate:
• Second derivatives w.r.t x and y are not equal Thus NOT a solution.

Step 3: Check option D.
\[ z = f(x^2 + y^2) \] Again:
• Second derivatives differ Thus NOT a solution.

Step 4: Final conclusion.
Non-solutions: \[ B, C, D \] Final Answer: \[ \boxed{B, C, D \text{ only}} \]