Numerical Methods for Engineers

Published by McGraw-Hill Science/Engineering/Math
ISBN 10: 0073401064
ISBN 13: 978-0-07340-106-5

Chapter 1 - Mathematical Modeling and Engineering Problem Solving - Problems - Page 24: 1.19a


\begin{equation} x(t)=\frac{g m}{c}\left(t+\frac{m}{c} e^{-\frac{c}{m} t}\right)-\frac{g m^{2}}{c^{2}} \end{equation}

Work Step by Step

\begin{equation} \begin{array}{l}{\text { Substituting }(1.10) \text { yields }} \\ {\qquad \frac{d x}{d t}=\frac{g m}{c}\left(1-e^{-\frac{c}{m} t}\right)} \\ {\text { The solution of this differential equation can be obtained by simply integrating both sides. }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\int d x=\int \frac{g m}{c}\left(1-e^{-\frac{c}{m} t}\right) d t} \\ {x(t)=\frac{g m}{c}\left(t+\frac{m}{c} e^{-\frac{c}{m} t}\right)+C} \\ {x(0)=0 \Rightarrow C=-\frac{g m^{2}}{c^{2}}} \\ {\Rightarrow x(t)=\frac{g m}{c}\left(t+\frac{m}{c} e^{-\frac{c}{m} t}\right)-\frac{g m^{2}}{c^{2}}}\end{array} \end{equation}
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