When $u ( x , y ) = X ( x ) Y ( y )$

question 8

Multiple Choice

When $u ( x , y ) = X ( x ) Y ( y )$ is substituted into the equation $u _ { xx } + u _ { y y } = 0$ , the resulting equations for $X$ and $Y$ are

• A) \(X ^ { \prime } + \lambda X = 0 , Y ^ { \prime } + \lambda Y = 0\)
• B) \(X ^ { \prime } + \lambda X = 0 , Y ^ { \prime } - \lambda Y = 0\)
• C) \(X ^ { \prime \prime } + \lambda X = 0 , Y ^ { \prime \prime } + \lambda Y = 0\)
• D) \(X ^ { \prime \prime } + \lambda X = 0 , Y ^ { \prime \prime } - \lambda Y = 0\)
• E) \(X ^ { \prime \prime } - \lambda X = 0 , Y ^ { \prime \prime } + \lambda Y = 0\)

---

Definitions:

Pollution

The introduction of contaminants into the natural environment, causing adverse changes and harm to ecosystems and human health.

Market Economy

A market mechanism where investment, production, and distribution decisions are informed by supply and demand forces, with the prices of goods and services set within an open pricing system.

Without Government

A scenario that describes a situation or theoretical condition where governmental institutions and interventions do not exist or are absent.

Pollution

The contamination of air, water, or soil by substances that are harmful to living organisms.