One solution of the spring equation with friction in Exercise 16 is x(t) = e t cos(t).

Question:

One solution of the spring equation with friction in Exercise 16 is x(t) = e^–t cos(t). Find v(t) and check that the solution matches the system of equations. What are the initial position and velocity?

The spring equation or simple harmonic oscillator

studied in Subsection 2.10.3 describes how acceleration (the second derivative of the position x) is equal to the negative of the position. The spring constant k has been set to 1 for simplicity. This one equation for the second derivative can be written as a system of two autonomous differential equations.