Trying to understand the transient current equation on a RLC circuit
This is really a question about solving differential equations. If you massage the first equation a bit you will see that the shape (the form of the function) for \$i\$, \$di/dt\$, and \$d^2i/dt^2\$ must all be the same. There are two such functions that we run into quite a bit in electronics: exponentials and sinusoids. For transient behavior the solution is often in the form of an exponential, so the author has substituted an exponential function for \$i\$. The remaining steps will be for solving for the constants \$A\$ and \$s\$.