Differential equation help

d²y/ dx² = - h y / j

where h, j are constants. What's y?

This on its own is not actually a question in my assignment but it is a starting point I need to get the problem done.

gabbagabbahey
Homework Helper
Gold Member
Can you think of a function whose second derivative is proportional to it?

I know y is an exponential function with a multiplication constant out the front, just not sure what the argument of the exponential is.

Actually I know it's going to be the sum of two exponentials with different multipliation contants and one will have the negative argument of the other.

In other words I know it's in the form

y = Aexp(c) + B exp (-c), just not sure what the c is.

gabbagabbahey
Homework Helper
Gold Member
Since, the derivative is respect to x, why not try $kx$ where $k$ is a constant (it may be complex) for your argument? In fact, the general solution has two terms $y(x)=Ae^{kx}+Be^{-kx}$. When you plug this into your DE, what do you get?

and k = sqrt (- h/j)?

so it is a complex argument as I was expecting because a complex exp can be written in terms of sines and cosines, whose 2nd derivative is their own negative.

gabbagabbahey
Homework Helper
Gold Member
Yes, exactly so you may as well write $y(x)=Csin(\frac{h}{j}x)+Dcos(\frac{h}{j}x)$

Thank you, gabbagabbahey.

This is a ay''+by'+cy=0 problem which has been discussed to death on every DE book. One should be able to write down the results (when b^2-4ac>0,<0 and = 0) while sleeping.