# Unit problem in differential equation

1. Jul 20, 2011

### sunipa.som

I have one differential equation like
dN/dt=c*(other terms with no unit)
unit of c is 1/ns. Now if I solve this equation, I will get value of N corresponding to t.
(1) Then what will be the unit of t?
(2) and if I calculate dN/dt1=(other terms with no unit). where t1=t/c.
Then what will be the unit of t1?

2. Jul 20, 2011

### hunt_mat

As I see it you have three possible units [N], [t] and [c], and I nonlimensionalise according to $N=N_{d}\bar{N}$, $t=t_{d}\bar{t}$ and $c=c_{d}\bar{c}$. There I put all the dimensions into the quantities with subscript d. So this turns the differential equation into:
$$\frac{d\bar{N}}{d\bar{t}}=\frac{c_{d}t_{d}}{N_{d}}\bar{c}$$
Where $c_{d}t_{d}/N_{d}$ is a nondimensional quantity. Does this clear things up?

3. Jul 20, 2011

### pmsrw3

t will be in ns. Your RHS has dimensions of time-1. dN/dt also has dimensions of time-1. So everything matches.

Are you sure you want to divide t by c? Maybe I'm confused, but I think it would make more sense to multiply t by c, in order to get a dimensionless version of the equation. For instance, if you define $\tau=ct$, then you get $\frac{dN}{d\tau}=\mbox{(other terms)}$ and c has gone away.

4. Jul 20, 2011

### sunipa.som

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Thank you. Sorry for mistake. I have to multiply t by c. Then I will get value of N for different times but then time has no unit.