Assume a bead of mass

m

slides along the curve

y = f ( x )

. Also assume that there is a damping force acting in the direction opposite to the velocity and proportional to the velocity, with proportionality constant

\beta

. The differential equation that describes the horizontal position of the bead is
A)

m \frac { d ^ { 2 } x } { d t ^ { 2 } } = - m g f ^ { \prime } ( x ) / \left( 1 - \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) - \beta \frac { d x } { d t }

m \frac { d ^ { 2 } x } { d t ^ { 2 } } = - m g f ^ { \prime } ( x ) / \left( 1 - \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) + \beta \frac { d x } { d t }

m \frac { d ^ { 2 } x } { d t ^ { 2 } } = - m g f ^ { \prime } ( x ) / \left( 1 + \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) - \beta \frac { d x } { d t }

m \frac { d ^ { 2 } x } { d t ^ { 2 } } = m g f ^ { \prime } ( x ) / \left( 1 + \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) - \beta \frac { d x } { d t }

m \frac { d ^ { 2 } x } { d t ^ { 2 } } = m g f ^ { \prime } ( x ) / \left( 1 + \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) + \beta \frac { d x } { d t }

The Differential Equation x′′−x′ex=0x ^ { \prime \prime } - x ^ { \prime } e ^ { x } = 0x′′−x′ex=0

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The Differential Equation $x ^ { \prime \prime } - x ^ { \prime } e ^ { x } = 0$

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