Remark. Equations of the form

with $a,b\in ℝ$, have wide-spread application throughout the sciences. Generally:

• the $y^{\text{'}\text{'}}$ term models inertia, the tendency of a system to maintain its current state

• the $a{y}^{\text{'}}$ term models dissipation, the transfer of energy to unresolved scales

• the $by$ term models storage, the transfer of energy to potential form

• The $f$ term models external forcing.

Remark. For $f=0$, (1) is a subcase of $y^{\text{'}\text{'}}+p\left(x\right)y^{\text{'}}+q\left(x\right)y=0$, hence the general solution is a linear combination $c_1{y}_1+c_2{y}_2$ of two independent solutions $y_1,y_2$ of (1) with $c_1,c_2\in ℝ$.

Remark. Guess a solution might be of form $y=e^{\lambda x}$, leading to the characteristic equation

with solutions

and distinct cases:

1. Two real roots if $a^2-4b>0$

2. Real double roots if $a^2-4b=0$

3. Complex conjugate roots if $a^2-4b<0$