Key Equations

Linear second-order differential equation	$a_{2} (x) y^{″} + a_{1} (x) y^{'} + a_{0} (x) y = r (x)$
Second-order equation with constant coefficients	$a y^{″} + b y^{'} + c y = 0$

Complementary equation	$a_{2} (x) y^{″} + a_{1} (x) y^{'} + a_{0} (x) y = 0$
General solution to a nonhomogeneous linear differential equation	$y (x) = c_{1} y_{1} (x) + c_{2} y_{2} (x) + y_{p} (x)$

Equation of simple harmonic motion	$x^{″} + ω^{2} x = 0$
Solution for simple harmonic motion	$x (t) = c_{1} cos (ω t) + c_{2} sin (ω t)$
Alternative form of solution for SHM	$x (t) = A sin (ω t + ϕ)$
Forced harmonic motion	$m x^{″} + b x^{'} + k x = f (t)$
Charge in a *RLC* series circuit	$L \frac{d^{2} q}{d t^{2}} + R \frac{d q}{d t} + \frac{1}{C} q = E (t)$