# Thread: Let n be the smallest digit in your K-Number that is greater than 1 and?

1. ## Let n be the smallest digit in your K-Number that is greater than 1 and?

Let n be the smallest digit in your K-Number that is greater than 1 and consider the ODE

dx/dt = (1/n) cos(t - nx)

Investigate the possibility of straight line solutions and discuss the asymp-
totic behaviour of x(t). (Your investigation and discussion must contain
rigorous mathematics )

2. Originally Posted by hazeleyes
Let n be the smallest digit in your K-Number that is greater than 1 and consider the ODE

dx/dt = (1/n) cos(t - nx)

Investigate the possibility of straight line solutions and discuss the asymp-
totic behaviour of x(t). (Your investigation and discussion must contain
rigorous mathematics )

I am assuming that you k-number is your student id number. I will use 5 in this example.

If we "guess" a linear solution then

$\displaystyle x(t)=mt+b \implies \frac{dx}{dt}=m$

If we put this into the ODE we get

$\displaystyle m=\frac{1}{5}\cos\left(t-5(mt+b) \right)$

First notice that the only way the trig function can be constant is if $\displaystyle m=\frac{1}{5}$ This gives

$\displaystyle \frac{1}{5}=\frac{1}{5}\cos\left(-5b \right)$

This will require

$\displaystyle -5b= \cos^{-1}(1) \iff -5b=2n\pi,n \in \mathbb{Z}$

3. Thank you so much!!! it helps a lot!!!
I'm wondering does it makes different if 'n' is an even number in the process? thanks

4. Originally Posted by hazeleyes
Thank you so much!!! it helps a lot!!!
I'm wondering does it makes different if 'n' is an even number in the process? thanks
n does not really have anything to do with the problem. If you replace 5 with n in my post and follow the same argument you will see that nothing changes!