Consider the following partial differential equation (PDE) ... where 𝑎 and 𝑏 are distinct positive real numbers. Select the combination(s) ...
Given PDE:
\(a\frac{\partial^2 f}{\partial x^2} + b\frac{\partial^2 f}{\partial y^2} = f(x,y)\)
Assume:
\(f=e^{\xi x+\eta y}\)
Step 1: Calculate derivatives
\[ \frac{\partial^2 f}{\partial x^2}=\xi^2 f \]
\[ \frac{\partial^2 f}{\partial y^2}=\eta^2 f \]
Step 2: Substitute into PDE
\[ a(\xi^2 f)+b(\eta^2 f)=f \]
Step 3: Take \(f\) common
\[ (a\xi^2+b\eta^2)f=f \]
Step 4: Cancel \(f\)
\[ \boxed{a\xi^2+b\eta^2=1} \]
Answer: Option A & B