The Boundary Value Problem $r \frac { d ^ { 2 } u } { d r ^ { 2 } } + 2 \frac { d u } { d r } = 0 , u ( a ) = u _ { 0 } , u ( b ) = u _ { 1 }$

The boundary value problem $r \frac { d ^ { 2 } u } { d r ^ { 2 } } + 2 \frac { d u } { d r } = 0 , u ( a ) = u _ { 0 } , u ( b ) = u _ { 1 }$ is a model for the temperature distribution between two concentric spheres of radii $a$ and $b$ , with $a < b$ .The solution of this problem is

A) $u = c _ { 2 } + c _ { 1 } / r \text {, where } c _ { 1 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a ) \text { and } c _ { 2 } = \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a )$
B) $u = c _ { 2 } + c _ { 1 } / r \text {, where } c _ { 1 } = \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a ) \text { and } c _ { 2 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a )$
C) $u = c _ { 2 } - c _ { 1 } / r , \text { where } c _ { 1 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a ) \text { and } c _ { 2 } = a b \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a )$
D) $u = c _ { 2 } - c _ { 1 } / r \text {, where } c _ { 1 } = \left( u _ { 1 } b - u _ { 0 } a \right) / ( b - a ) \text { and } c _ { 2 } = a b \left( u _ { 1 } - u _ { 0 } \right) / ( b - a )$
E) none of the above