# Help toward solving second order non-linear differential equation

1. Jul 22, 2010

### manikandanb

Hi,

I have a differential equation of the form
d2 x
---------------- = g/z * x(t)
dt2

Here g and z are constants. So, this is a 2nd order ODE which has a closed form solution.
In fact, i know the solution for x in terms of cosh and sinh functions.

In the above differential equation, if z is not constant. That is, id the the eqn becomes
d2 x
---------------- = g/z(t) * x(t)
dt2

Is it still correct to assume x is only a function of time?
If z(t) is sinusoidal, then this equation is a non-linear 2nd order ODE. Am i right?

Please point me to a standard textbook which might help me solve these kind of non-linear diff.equations? Any suggestions/pointers are welcome. Thank You in advance.

--MB

2. Jul 22, 2010

### Office_Shredder

Staff Emeritus
It's still a linear differential equation. If x and y are both solutions

$$\frac{d^2(x+y)}{dt^2}=\frac{d^2x}{dt^2}+\frac{d^2y}{dt^2}=\frac{g}{z(t)}x + \frac{g}{z(t)}y = \frac{g}{z(t)}(x+y)$$

You can also show that scalar multiples are solutions