# Math Help - Help interpreting these solutions

1. ## Help interpreting these solutions

http://i.imgur.com/seDNI8K.jpg

for question 1,
does the solution approach the equilibrium faster in(b) that it does (a) because of the fact that it e^-2t as opposed to e^-t ?
What does that mean precisely?

Also, why is it that the solution approaches the equilibrium in the first section, but it diverges in the answers to question 2?

thanks.

2. ## Re: Help interpreting these solutions

nvm, solved.

dy/dt + sin(t+y) = sin(t)
is non-linear? I thought a function was only non-linear if it the independent variable (in this instance, y) was not to the first power?

3. ## Re: Help interpreting these solutions

Originally Posted by 99.95
http://i.imgur.com/seDNI8K.jpg

for question 1,
does the solution approach the equilibrium faster in(b) that it does (a) because of the fact that it e^-2t as opposed to e^-t ?
What does that mean precisely?
Did you check some numerical values? Taking, say, $y_0= 0$, we have $5(1- e^{-t})$, $(5/2)(1- e^{-2t})$, and $5(1- e^{-2t})$. When t= 0, they are, of course, all equal to 0. The distance from the equilibrium values, 5, 5/2, and 5, are 5, 5/2, and 5. When t= 1, they are 5(1- 1/e)= 3.1606, (5/2)(1- 1/e^2)= 2.1617, and 5(1- 1/e^2)= 4.323. When x= 2, they are 5(1- 1/e^2)= 4.323, (5/2)(1- 1/e^4)= 2.4542, and 5(1- 1/e^4)= 4.908. Do you see that the terms with "2t" in the exponent are getting close to the equilibrium values faster?

Also, why is it that the solution approaches the equilibrium in the first section, but it diverges in the answers to question 2?

thanks.
The answers in question 2 have "t" and "2t" in the exponent rather than "-t" and "-2t". As t increases, those terms will increase rather than decrease.