In the previous two problems, the product solutions are
A) $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$
B) $\cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$
C) $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) / L }$
D) $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } x ^ { 2 } t / L ^ { 2 } }$
E) $\cos ( ( n - 1 / 2 ) \pi x / L ) e ^ { ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$

Question

Quizplus · Accepted Answer

The correct choice is A because it represents a solution to the heat equation or wave equation in a one-dimensional domain, which typically involves a sine function for spatial variation and an exponential decay in time, indicating diffusion or damping. The presence of $\pi$ and $L$ in both the sine and exponential parts, along with the specific form of the decay term, is characteristic of solutions to these types of partial differential equations.

In the Previous Two Problems, the Product Solutions Are A) sin⁡((n−1/2)πx/L)e−(n−1/2)2π2t/L2\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }sin((n−1/2)πx/L)e−(n−1/2)2π2t/L2

In the Previous Two Problems, the Product Solutions Are
A) $\sin ( ( n - 1 / 2 ) \pi x / L ) e ^ { - ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } t / L ^ { 2 } }$