# Math Help - Separation of variables

1. ## Separation of variables

Hi, I need help on this problem

Suppose that a certain population has a growth rate that varies with time and that this population satisfies the differential equation

dy/dt = (0.5 + sin t)y/5

If y(0)=1, find or estimate the time at which the population has doubled.

I think that this can be done with separation of variables. If we cross multiply then we get 5 dy (1/y) = dt (0.5 + sint)
Then integrate:

5 ln|y| = 0.5t - cost

Does this seem correct?

2. Originally Posted by RB06
Hi, I need help on this problem

Suppose that a certain population has a growth rate that varies with time and that this population satisfies the differential equation

dy/dt = (0.5 + sin t)y/5

If y(0)=1, find or estimate the time at which the population has doubled.

I think that this can be done with separation of variables. If we cross multiply then we get 5 dy (1/y) = dt (0.5 + sint)
Then integrate:

5 ln|y| = 0.5t - cost

Does this seem correct?
You seem ok so far except you forgot the arbitrary constant on the RHS. So,
5 ln|y| = 0.5t - cost + C
=> ln|y| = 0.1t - 0.2cost + C
=> y = e^(0.1t - 0.2cost + C)
Now y(0) = 1
=> 1 = e^(0.1(0) - 0.2cos(0) + C)
=> 0.1(0) - 0.2cos(0) + C = 0
=> - 0.2 + C = 0
=> C = 0.2

so we have y = e^(0.1t - 0.2cost + 0.2)