In this video lecture, Prof. Gilbert Strang discusses Differential Equations of Growth.
The key model for growth (or decay when c < 0) is dy/dt = c y(t). The next model allows a steady source (constant s in dy/dt = cy + s). The solutions include an exponential e^ct (because its derivative brings down c). So growth forever if c is positive and decay if c is negative. A neat model for the population P(t) adds in minus sP^2 (so P won't grow forever). This is nonlinear but luckily the equation for y = 1/P is linear and we solve it.
Population P follows an "S-curve" reaching a number like 10 or 11 billion. Great lecture but Professor Strang should have written e^-ct in the last formula
Highlights of Calculus is a series of short videos that introduces the basics of calculus—how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.
The series is divided into three sections: Introduction - Why Professor Strang created these videos - How to use the materials
Highlights of Calculus - Five videos reviewing the key topics and ideas of calculus - Applications to real-life situations and problems - Additional summary slides and practice problems
Derivatives - Twelve videos focused on differential calculus - More applications to real-life situations and problems - Additional summary slides and practice problems
Acknowledgements Special thanks to Professor J.C. Nave for his help and advice on the development and recording of this program.The video editing was funded by the Lord Foundation of Massachusetts.