In the three previous problems, the product solutions are
A) $u _ { n } = \left( a _ { n } r ^ { n } + b _ { n } r ^ { - n } \right) \left( c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) \right) , n = 0,1,2 , \ldots$
B) $u _ { n } = r ^ { n } \left( c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) \right) , n = 1,2,3 , \ldots$
C) $u _ { n } = r ^ { n } \left( c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) \right) , n = 0,1,2 , \ldots$
D) $u _ { n } = r ^ { n } \left( c _ { n } e ^ { n \theta } + d _ { n } e ^ { - n \theta } \right) , n = 0,1,2 , \ldots$
E) $u _ { n } = r ^ { n } \left( c _ { n } e ^ { n \theta } + d _ { n } e ^ { - n \theta } \right) , n = 1,2,3 , \ldots$

Question

Quizplus · Accepted Answer

Option C correctly represents a general solution for a series of problems involving radial and angular components, where $r^n$ indicates a radial dependence that scales with $n$, and the combination of $\cos(n\theta)$ and $\sin(n\theta)$ reflects the angular dependence. The inclusion of $n = 0,1,2, \ldots$ allows for a constant term in the series, which is often necessary in solutions to physical problems, such as those governed by the Laplace or Helmholtz equations in polar coordinates.

In the Three Previous Problems, the Product Solutions Are A)

In the Three Previous Problems, the Product Solutions Are
A)