Second order homogenous differential equation I need to find a solution to the equation:
$$m(t)x''(t) + kx(t) = 0$$
This is similar to the equation of oscillation of a spring, with one difference, that m(t) is not constant in time. Also, m(t) is not given by any analytic formula, though we can assume that it is continuous, differentiable and I know m(t) for any t. Because of that, I do not know how to apply the principle of superposition.
 A: One cannot answer on a general manner. The results can be very divers depending on the kind of function $m(t)$. For examples ( non exhaustive of course) :
Let : $$\frac{x''(t)}{x(t)} = -\frac{k}{m(t)} = \Psi(t)$$
$\Psi(t)=a \quad\to\quad x(t)$ involves $\exp$ , $\cosh$ , $\sinh$ , $\cos$, $\sin$ functions.
$\Psi(t)=at+b \quad\to\quad x(t)$ involves Airy functions.
$\Psi(t)=at^2+bt+c \quad\to\quad x(t)$ involves Parabolic Cylinder functions.
$\Psi(t)=at^b \quad\to\quad x(t)$ involves Bessel functions.
$\Psi(t)=ae^{bt}+c \quad\to\quad x(t)$ involves Bessel functions.
$\Psi(t)=\frac{a}{x+b} \quad\to\quad x(t)$ involves Bessel functions.
$\Psi(t)=\frac{a}{t^2}+\frac{b}{t}+c \quad\to\quad x(t)$ involves Confluent Hypergeometric and/or Kummer and/or Whittaker functions.
And so on...
Many others involves Hypergeometric functions to express $x(t)$ on closed form.
But more generally, in many cases one cannot express the solution on closed form because a convenient special function is not available.
